Archive for the ‘inclusive practices’ Category

2018-19 primary teachers study group: session 4

Posted on: March 11th, 2019 by jnovakowski

Our fourth session was held at Blair Elementary, hosted by Karen and Tanyia. They shared the development of their outdoor learning space and how it and the gardens are being used by teachers and students in the school.

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We walked around the school grounds, looking for inspiration for mathematical thinking. At this time of year, you can really see the structure of the deciduous trees and it is an opportunity to notice lines, shapes and angles. With moss and lichen growing on some trees and on fences, there are lots of math-inspired questions that can be investigated around the life cycle, size and growth of these unique living things.

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We noticed many trees with some interesting growth patterns and markings (some caused by pruning according to our master gardener Megan). What stories live in these trees? What might a timeline of a tree’s life look like? Seasons, years, decades – such an interesting lens to explore concepts of time through.

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Sarah Regan and Megan Zeni were awarded this year’s June Chiba Sabbatical and used their release time to visit several outdoor/nature focused schools across southern BC. We were happy to host them and have them share some of their experiences.

IMG_8871Our next study group book is the Canadian children’s book Flow Spin Grow: Looking for Patterns in Nature. It connects really well with our focus this year of finding and investigating mathematics outdoors. After sharing our focus on twitter, the author shared his website where he has curated some resources to complement the book HERE.

IMG_8882 I know my eyes will be open for all sorts of patterns – branching, spiralling, spinning – as spring emerges around us.

Have a lovely spring break!

~Janice

 

 

 

2018-19 primary teachers study group: session 3

Posted on: March 11th, 2019 by jnovakowski

Our third session was hosted by Jessica, Lisa, Laura and Sasha at Anderson Elementary. A couple of the teachers are particularly knowledgable about mushrooms and shared information about the fungi in the neighbourhood.

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We walked through Garden City Park and the Anderson teachers shared how they use the space over the school year to observe and document seasonal changes. The students were also very observant of how the windstorms this fall/winter affected the park and the changes created by the storms. The City of Richmond has created an arboretum area on the west side of the park, with plaques identifying and describing the trees. There is also the opportunity to observe birds and other urban wildlife in the park.

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A focus of some of the teachers has been on the human impact on the local environment. Some questions for teachers and students to consider:

How can mathematics help us to understand this issue?

What data/information could be collected and how could it be shared?

What information could be collected?

How might different ways of sharing information have an impact on understanding of the issue?

What actions could we take?

 

Looking forward to seeing how different schools and classes make connections between mathematics and their outdoor environment.

~Janice

February thinking together: develop, use and apply multiple strategies to solve problems

Posted on: February 28th, 2019 by jnovakowski

This month’s curricular competency focus is using multiple strategies to solve problems. There is a development in how strategies are used from K-12 and for what types of problems.

In K-5 the curricular competency language is “develop and use multiple strategies to engage in problem solving” with elaborations including examples of strategies involving visual, oral and symbolic forms and through play and experimentation.

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In K-5, we support students in developing a repertoire of strategies to draw upon and we encourage the practice of choosing and using these strategies in different problem solving experiences ranging  from structured word/story problems, open problems or questions or problem-based or numeracy tasks. During the development of strategies, students will notice similar strategies being shared by their classmates and these strategies might be named such as “looking for a pattern” or “acting it out” or “represent with materials”. Naming strategies such as these helps to enhance mathematical communication, discourse and community in the classroom when discussing mathematical problems.

As with many of the curricular competencies in math, there are slight variations between grade bands, showing the developing application and demonstration of these competencies.

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In grades 6-9 the curricular competency language is “apply multiple strategies to solve problems in both abstract and contextualized situations” with elaborations including examples of strategies focusing on those that are familiar, personal or from other cultures. Students in this grade range are refining and reflecting on their own use of problem solving strategies and we encourage students to listen and learn from their peers in order to consider new ways to think about a mathematics problem.

 

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In grade 10 the curricular competency language is “apply flexible and strategic approaches to solve problems” with elaborations such as deciding what tools to use to solve a problem as choosing from a list of known strategies such as guess and check, solve a simpler problem, model, use a chart, role-play or use diagrams. The numeracy processes for engaging in numeracy tasks are related to this competency at the secondary level - interpret, apply, solve, analyze and communicate.

 

Although specific strategies such as “guess and check” or “solve a simpler problem” are not named specifically in the elaborations from K-9, it is these more formally named strategies that are developed with understanding, meaning and purpose over time. Alternative or personally derived or preferred strategies may also be developed by students and shared with their solutions, supported with their reasoning and explanations to demonstrate their understanding of the problem and the mathematics involved.

Many math educators and researchers have found over decades of research and classroom experiences that students who have multiple strategies or approaches to problems are more fluent and flexible in their thinking. An important aspect of using multiple strategies is knowing when a particularly strategy is helpful or efficient. Not all strategies are suitable for all problems and this an important part of the progression of developing this competency in mathematics  One particularly effective instructional strategy is engaging students in comparing the strategies they used to solve a problem. Researchers have recently examined the cognitive process of comparison and how it supports learning in mathematics. The sharing and comparison of multiple student strategies for a problem was found to be particularly effective for developing procedural flexibility across students and to support conceptual and procedural knowledge for students with some background knowledge around one of the strategies compared. (Durkin et al, 2017 – referenced below). Based on their findings, the researchers share some significant instructional moves that will support student learning:

1) regular and frequent comparison of  alternative strategies

2) judicious selection of strategies and problems to compare

3) carefully designed visual presentation of the multiple strategies

4) small group and whole class discussions around comparison of strategies with a focus on similarities, differences, affordances and constraints

 

Examples of what the use of multiple strategies might look like in the classroom include:

Primary: The teacher reads the story The Frog in the Bog and asks the grade 1 students to figure out how many critters are in the frog’s tummy. The teacher invites the students to think about how they might solve this problem and what they will need. The students work on their own or with a partner to solve the problem through building with materials, acting it out, drawing or recording with tally marks and numbers. Some students accompany their solutions with an equation and one student records his ideas orally using iPad technology. As the students are working, the teacher pauses the students and asks them to walk around the room and see what their classmates are doing and see if they can find a new idea for their own work. After solving the problem, the students prepare to share their solutions and strategies with the class and the teacher gathers the students on the carpet and chooses some students who used different strategies to share. The teacher records the strategies on the chart and then asks the students if they have a new idea for a strategy for the next time they do a problem like this.

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Intermediate: In a grades 6&7 class, the teacher projects the first three figures of a visual pattern on the class whiteboard (examples on visual patterns.org). The teacher asks the students what they notice about the figures and records some of the students’ responses and then asks them to consider what comes next. Students are asked to consider what strategies or approaches might help them think about this. After some thinking time, the teacher asks the students to turn and talk with one or two other students and compare each others’ strategies and consider new ways of thinking about the problem. The teacher then invites the students to apply more than one strategy to solve what figure 43 will look like. The students share their solutions and strategies with the teacher recording the different strategies through different representations such as a drawing, a narrative, an expression, a table or a graph. The teacher then facilitates a discussion comparing the representations and how they are connected and support the understanding of the problem.

(with thanks to Fawn Nguyen and Marc Garneau for the inspiration)

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Secondary: Students in a grade 10 class are assigned to be in random groups of three and work on a numeracy task on a whiteboard or window around the classroom. The class has been learning about prime factorization and the teacher shares the following problem orally:

Prime numbers have exactly two factors – 1 and itself. Which numbers have exactly 3 factors? Exactly 4 factors? And so on. Given any positive integer, n, how can you tell exactly how many factors it has?

Each group of students begins talking and sharing their ideas. Students begin to record their thinking, using diagrams, charts, numbers, etc. and build on and challenge each others’ thinking about the problem and approaches to solving it. Students move around the room and watch or engage with other groups. The teacher facilitates students’ sharing of solutions and approaches to the problem and then provides a set of related problems for students to continue practicing with, either in their groups or independently.

Numeracy tasks such as this one can be found HERE and HERE and HERE.

(with thanks to Mike Pruner and Dr. Peter Liljedahl for the thinking classroom inspiration)

 

Some questions to consider as you plan for learning opportunities to develop the competency of using multiple strategies and approaches to solve problems:

What strategies and approaches do you notice your students using? Are some students “stuck” using the same strategy? How could you nudge students to try different strategies and approaches?

What different types and structures of math problems are being provided to your students? Are students flexible with their strategy choice or approach, making decisions based on the problem they are working on?

How might you and your students record their strategies and approaches to make this thinking visible?

What opportunities are we creating for students to watch and listen to others think through, choose and apply strategies and solve problems? How might this support their learning?

What tools, materials and resources do students have access to to support choice and application of different strategies and approaches when solving math problems?

~Janice

References

Elementary and Middle School Mathematics: Teaching Developmentally by John van de Walle et al

Teaching Mathematics through Problem-Solving (NCTM) edited by Frank Lester and Randall Charles

Why Is Teaching With Problem Solving Important to Student Learning (NCTM Research Brief)

Durkin, K., Star, Jon. R. & Rittle-Johnson, B. (2017) Using Comparison of Multiple Strategies in the Mathematics Classroom: Lessons Learned and Next Steps, ZDM: The International Journal on Matheamtics Education 49(4), 585-597.

 

January thinking together: use technology to explore mathematics

Posted on: January 31st, 2019 by jnovakowski

This month’s focus is on the curricular competency: use technology to explore mathematics.

This is the language that is used from K-5 with the accompanying elaborations:

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This is the language of the learning standard for grades 6-9:

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And this is the language of the learning standard in grades 10-12, with elaborations that are more course-specific:

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There are many questions that arise for educators and parents around the use of technology. In some contexts the use of personal devices becomes a management and liability concern for schools and in other contexts there are access and equity concerns around technology. In terms of pedagogy and appropriate use, there is always a professional judgement made as to the suitable use of technology and whether it is enhancing the learning experience in some way. Technology is not to be used just for the sake of using technology but instead, choices are made around technology use based on intention, context and purpose. In mathematics, there are many applications that allow for students to visualize and experience mathematics in ways they would not otherwise be able to (one example is the use of Desmos). Another aspect of using technology in mathematics teaching is as a tool to represent and share students’ learning. There are many accessibility features available on devices for students who may need different tools to support their communication or recording of ideas. Technology can be a powerful tool to support inclusive practices, choice and differentiation.

When we look at BC’s redesigned curriculum for information on the role of technology within a learning environment, the following is shared:

ICT-enabled learning environments

Students need opportunities to develop the competencies required to use current and emerging technologies effectively in all aspects of their learning and life. Technology can facilitate collaboration between students, educators, parents, and classrooms while also providing schools with rich online resources. Today’s technology enables classrooms, communities, and experts around the world to share digitally in a learning experience, wherever they may be.

source: https://curriculum.gov.bc.ca/curriculum/overview

E-Portfolios

Communication with families (and others) is an important part of our education system and in our district we are embracing e-portfolios and the use of technology to share and communicate student learning and progress with families. Students are able to take photographs or videos and upload them to their portfolios and annotate their posts with information or self-assessment about their learning. The teacher is also able to add descriptive feedback that is shared between teacher, student and family.

Screencasting

As a classroom and resource teacher and teacher-librarian, one of my favourite uses of technology was the use of screen casting apps. These apps allow students to take a photograph of the math they have been building, creating, diagramming or recording and then use annotation tools such as text labelling and arrows to explain their thinking as well as using audio tools to narrate their thinking. I found that many students were more confident and detailed in sharing their learning through these apps that what I might have found out about their understanding in other ways. There is also an honouring of students’ uniqueness in how they might see or think through the mathematics that can be shown through these types of apps. Some examples of screen casting apps we use in our district our: ShowMe, Educreations, Explain Everything, 30Hands and Doceri.

Math Apps

There are many apps that can support mathematics learning – some are mathematics specific and others are used to represent and share learning. A caution is the type of math apps that are essentially a worksheet and don’t include any sort of feedback to students, visual supports, problem-solving or mathematical thinking. Some locally produced apps include the TouchCounts from SFU that uses the research around gesturing to create an interactive app that focuses on counting and decomposition and composition of quantities. Another series of BC apps are the MathTappers apps developed through the University of Victoria. Each app has visual supports for students developing their understanding of a concept as well as symbolic or abstract notation. There are also choices as the number range that students can work with, allowing for differentiation. These apps are all on our district configured iPad devices. Some specific apps from this series include Find Sums, Multiples, and Equivalents.

Screen Shot 2019-01-31 at 11.10.00 AMThe apps from the Math Learning Centre are also on our district configured iPad devices and allow for content creation and capturing students’ process and thinking. These apps are in web-based and iOS and Android formats. More information can be found HERE.

 

 

There are also so many apps that allow for students to share their thinking such as ShowMe, Educreations, Book Creator, PicCollage, 30Hands and Doceri.

Tracy Zager shares her ideas on evaluating math content apps HERE. Her non-negotiable criteria are:

1) no time pressure

2) conceptual basis for operations

3) mistakes are handled productively

Read through her blog post for explanation and examples.

The following is a link to some recommended apps and blog posts about students using them from #summertech15 and HERE is a blog post about using iPad technology and specific apps to support all students in mathematics.

 

Calculators

Although BC does not yet have a specific statement on calculator use, there is no intent that students will use calculators to complete calculations instead of learning the concepts and practice involved with operations (addition, subtraction, multiplication, division). In some cases, students that have specific learning needs and plans may use calculators as an adaptation. In some cases, teachers may choose to provide the choice of calculators when the focus of the lesson or assessment is not on calculation but on another area of the math curriculum such as problem-solving and calculators can be used for the necessary calculations so that students can focus on the other aspects of the task. Calculators can also be used to investigate patterns and relationships, support student reasoning or justification.

The NCTM has a research brief on calculator use in the classroom which can be found HERE as well as a position paper on calculator use in elementary grades which can be found HERE.

Virtual Manipulatives

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The Math Learning Centre offers a variety of virtual manipulatives in web, iOS and Android formats. They can be accessed HERE.

 

 

 

desmos

Desmos is a free, online graphing calculator application that is used by teachers and students all over the world. There are both web-based and app platforms. Students are “able” to play with parameters in an equation and visually see how the graph changes as the parameters change.  The desmos staff and teachers across the world have developed lessons and tasks that are open source and shared through the desmos teacher website at no cost HERE. There is also an activity builder so that teachers can create their own tasks.

I attended a math conference a few years ago where Eli Luberoff, CEO of desmos, shared his passion for the teaching and learning enabled and enhanced by this tool. In particular, I was captivated by the marble slides task he shared and the authentic learning that we witnessed happening for students in the video he shared.

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More information about Desmos and access to many classroom activities can be found HERE.

Coding and Computational Thinking

There are many links between coding and computational thinking. Two new senior math courses – Computer Science 11 and 12 have been added to our BC curriculum framework and these courses focus on coding, programming and computational thinking.  I will be sharing a blog post specific to coding and math in the next few months.

Osmo

Screen Shot 2019-01-29 at 10.56.25 PMOsmo is an interactive accessory for iPad technology that uses the camera to create Reflective Artificial Intelligence. The red camera clip and white base are used with free apps and game materials that can be purchased online or at the Apple Store. Two of its earliest games focused on mathematics – the Tangram game focuses on spatial reasoning and the Numbers games focuses on decomposition and composition of numbers. Osmo is always developing new games including a Pizza game that focuses on financial literacy and a series of coding games.

More information about Osmo can be found in a blog post here and on their website here. The SD38 DRC has five Osmo kits available to borrow. Note that one iPad device is needed for each kit.

Augmented Reality (AR)

Augmented reality (AR) is an interactive experience of a real-world environment where the objects that reside in the real-world are “augmented” by computer-generated perceptual information, sometimes across multiple sensory modalities (from Wikipedia). There is an interplay in AR between digital and real-world environments whereas in Virtual Reality (VR) you engage with a simulated environment. A few years ago we had a Google Expeditions team visit Homma school and share their Google cardboard virtual reality devices with the students. A blog post about that experience can be found HERE. This was a first foray into thinking about ways this kind of technology could support teaching and learning. My first experience with AR was a few years ago when the colAR app created a special event to go along with Dot Day (inspired by the book by Peter Reynolds). The information about this can be found HERE and is a great starting point to use AR with students.

Our new technology integration teacher consultant Ellen Reid has been exploring AR with the iPad app AR Maker . We talked about the mathematical possibilities for using AR and along with the development of spatial reasoning, the following concepts came to mind: surface area, volume, transformational geometry, scale, proportion, ratio, 2D and 3D geometry, and composition and decomposition of shapes. The following are some photos Ellen captured as she created AR WODBs (Which One Doesn’t Belong?):

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For Richmond teachers, please also check out the Integrating Technology for Teachers page, curated by Chris Loat, on our district portal linked HERE.

 

Some questions to consider as you plan for learning opportunities to develop the competency of using technology to explore mathematics:

How can technology enhance students’ mathematical experience and see and think about mathematics in different ways?

What specific curricular content and competencies at your grade level could be explored and investigated through technology, including the use of calculators?

How can technology be used to support students’ collaboration and communication in mathematics?

What opportunities are we creating for sharing and communication with families through the use of technology?  How are we communicating with parents how forms of technology are being used in our schools to support learning in mathematics?

~Janice

big mathematical ideas for K-2 2018

Posted on: December 19th, 2018 by jnovakowski

This fall we hosted a three-part after school professional learning series focusing on the big mathematical ideas in Kindergarten thru Grade 2. We have been doing this series for grades 3-5 teachers for the last five years and this year have added series for K-2 and grades 6-9 teachers. The focus of the series is to look at the foundational math concepts within the grade band and consider ways to develop those concepts and related curricular competencies. Other curricular elements such as core competencies, First Peoples Principles of Learning, use of technology and assessment are woven into the series.

September 27

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We discussed three instructional routines focused on counting: choral counting, count around the circle and counting collections. The following are the professional resources that were recommended and every teacher attending was provided with a copy of Christopher Danielson’s new book How Many? and the accompanying teachers guide.

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We shared the idea of unit chats which is the essence of the book How Many? What could we count? What else could we count? How does the quantity change as we change the unit we are counting?

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We also introduced Dan Finkel’s website and his section of photographs that can be used for unit chats HERE.

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Between the first and second sessions, teachers were asked to try one of the counting routines, read parts of the How Many? teacher guide, try a unit chat with their classes and do the performance task with one of their students.

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October 25

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We spent the first part of our session together sharing with each other about a counting routine they did with their class, how their students responded to unit chats and their findings from the performance task. Teachers brought video, photos and student work to share and discuss.

We discussed the importance of research-based learning trajectories/progressions to inform our instructional and assessment practices. The BC Numeracy Network has collated several learning trajectories/progressions HERE (scroll down to the bottom of this page).

We introduced the draft of the new SD38 Early Numeracy Assessment Tool which is intended to use with students from the end of Kindergarten through grade 2 to create class learning profiles and well as help identify specific learning goals for students. It can also be used by schools to monitor student progress over time. The assessment tool focuses on key areas of number sense and the tasks are drawn from the BC Early Numeracy Project and the work from the Numerical Cognition Lab at Western University. Teachers were asked to complete the assessment with one student they were curious about learning more about.

November 22

We began our session sharing how it went with the new K-2 assessment tool. The teachers had lots of good feedback and suggested edits which will now be taken back to the district committee for final revisions.

We shared some different materials and experiences to support the development of K-2 students’ number sense, connecting the ideas of counting, subitizing, connecting quantities and symbols and ordering/sequencing. One of our favourite materials is Tiny Polka Dot, which I personally believe should be in every K-2 classroom (available in Canada through amazon.ca HERE).

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We also went over the ten frame games and tasks that can be used in K-2 classrooms for purposeful practice during math workshop or small group instructional time.

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Teachers and their students took photographs to contribute to our own digital How Many? book and it is a work in progress but the collection we have so far can be found here (best viewed via Chrome):

How Many? digital book

Look for information and  next steps for our SD38 K-2 Numeracy Assessment Tool in the new year!

~Janice

 

 

 

 

 

 

creating spaces for playful inquiry: encounters with charcoal

Posted on: December 14th, 2018 by jnovakowski

To launch the 2018-19 season of our ongoing professional learning series, Creating Spaces for Playful Inquiry, we created opportunities for educators to have encounters with charcoal and make connections to teaching and learning across the BC curriculum. Inspired by our learning from Opal School in Portland to use different materials to explore ideas and emotions through an aesthetic dimension, we chose charcoal specifically as we believed it was a material that educators might need some support with, in understanding the material in new ways.

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We shared a blog post from the Opal School Blog: Thinking with Charcoal

and shared the Canadian books The Art of Land-Based Early Learning (volumes 1 and 2) that can be found HERE.

I actually experimented with making my own charcoal. I trimmed some willow branches from my backyard, tightly wrapped them in cheesecloth and then aluminum foil (to eliminate any oxygen inside) and put them in our fire pit. I didn’t have enough wood to maintain a high enough heat for long enough (researched needing about an hour) so I “finished” the packages the barbecue. They worked out quite well but next time, I will strip the bark off the twigs first.

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We curated a collection of charcoal and related materials from DeSerres and Phoenix Art Studio

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and invited educators to engage with materials, ideas and concepts.

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Our resource document about charcoal, including the questions provided to provoke educators’ thinking can be found here:

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Some educators commented that it was their very first time using charcoal themselves and they reflected on what it meant to explore a material for the first time, how that made them feel both curious and vulnerable and also sparked many connections and ideas for using charcoal with their students.

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Two of our playful inquiry mentors, Sharon and Christy, shared experiences and stories from their classrooms

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and then after dinner together, we broke off into mentor group to share ideas and think together about ways to engage with playful inquiry this school year.

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We have been growing our playful inquiry community in our district for several years now with both our own initiatives and projects as well as continuing to nurture our relationship with Opal School and it is exciting to continue to welcome teachers into our conversations. Our next district event will be an open studio at the district conference on February 15 and a playful inquiry symposium on the afternoon of the district pro-d day on May 17.

 

~Janice, on behalf of the playful inquiry mentors

 

 

2018-19 primary teachers study group: session 2

Posted on: December 12th, 2018 by jnovakowski

Our second session of this year’s primary teachers study group was hosted by Anna and Shannon at McNeely Elementary. Anna shared the book about mushrooms that her students researched and wrote after finding and investigating the mushrooms they found in their mini-forest near the school.

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 The class was also inspired by one of our study group books, Anywhere Artist, and went out into their mini-forest to create art with found materials.

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The land art of UK artist James Brunt (on twitter at @RFJamesUK) also inspired us to take on the #100LeavesChallenge.

Anna and Shannon toured us through McNeely’s new outdoor learning space and through their mini-forest, adjacent to the school.

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Together we shared ideas for how different plants, trees and animals could inspire mathematical thinking or questions to investigate.

Thank you to Anna and Shannon for hosting us!

~Janice

2018-19 primary teachers study group: session 1

Posted on: December 12th, 2018 by jnovakowski

Beginning our sixteenth year, the Richmond Primary Teachers Study Group met for the first time this school year on October 11 at Diefenbaker Elementary. As agreed upon by study group participants, this year’s focus is on the teaching and learning of mathematics in places and spaces outdoors, considering both how to take mathematics outdoors but also how the outdoors can inspire mathematical thinking.

Our three study groups books that we are going to draw inspiration from this year are:

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Messy Maths by Juliet Robertson

50 Fantastic Ideas for Maths Outdoors by Kristine Beeley

Anywhere Artist by Nikki Slade Robinson

 

There are so many books and resources available to support our professional inquiry together this year.

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We spent some time exploring the Diefenbaker garden, playground and new outdoor learning area and considering what math we could find in these spaces.

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IMG_3270One of the tasks we did was using materials or referents to estimate and create the length of one metre. We followed this up by each making our own “Sammy the Snake” – a one metre length of rope (idea from the Messy Maths book). This length of rope can be part of a “go bag” to take outside for measuring lengths, perimeter, circumference of trees and to think about fractions (by folding the length of rope). It is a flexible tool to support students’ developing understanding of comparing, ordering and constructing concepts of measurement and number.

 

 

 

 

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Thanks to the Diefenbaker team for hosting us!

~Janice

 

December thinking together: visualize to explore mathematical concepts

Posted on: December 11th, 2018 by jnovakowski

This month’s focus is on the curricular competency: visualize to explore mathematical concepts.

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In the 2007 WNCP mathematics curriculum, visualization is defined as involving “thinking in pictures and images, and the ability to perceive, transform and recreate different aspects of the visual-spatial world”. Concepts such as number, spatial relationships, linear relationships, measurement, and functions and relations can be explored and developed through visualization.

In the new BC grades 10-12 courses, the elaborations for this curricular competency are:

  • create and use mental images to support understanding
  • visualization can be supported using dynamic materials (e.g., graphical relationships and simulations), concrete materials, drawings, and diagrams

Visualization and spatial reasoning involve the relationship between 2D and 3D shapes as well as dynamic imagery such as different perspectives, movement, rotations and reflections. Visualizing involves an interplay between internal imagery and external representations  (Crapo cited in NRICH article below). Students need experience with concrete and visual representations/pictures/models as well as being able to visualize something in their minds, often referred to as the “mind’s eye”.

Canadian and International research has shown that there are links between strong abilities to visualize and success in mathematics. One widely used psychological assessment for visualization involves “The Paper Folding Test”  in which a paper is folded and a hole is placed through a specific location and the participant is asked to visualize what the paper will look like when it is unfolded, utilizing the ability to generate, maintain and manipulate a mental image, (Lohman, 1996 cited in Moss et al 2016). A recent study also found a link between the ability to visualize and success with solving mathematical word problems, citing the ability to mentally visualize and make sense of the problem contributed to success in diagramming and solving problems (Boonen et al 2013 cited in Moss et al 2016). The Canadian work of (Moss et al 2016 ) and their Math for Young Children research project focuses on spatial reasoning and the importance of developing students’ flexible use of visualization skills and strategies.

 

Instructional Resources

Screen Shot 2018-12-11 at 4.11.50 PMThe book Taking Shape (referenced below) provides several visualization tasks on pages 30-35 but visualization is an important component of most of the spatial reasoning tasks in the book.

 

 

 

 

Quick Images is an instructional routine that supports the visualization of quantities and shapes. Dot patterns and Screen Shot 2018-12-11 at 2.26.05 PMcomposition of shapes are often used as quick images. More information and videos can be found on the TEDD website HERE.

 

A short article from the NCTM explaining the connection between visualization and subitizing can be found here:

NCTM_quickimages_tcm2016-12-320a

 

Screen Shot 2018-12-11 at 2.28.51 PMFawn Nguyen has compiled a collection of visual patterns HERE. Visual patterns provide the first three steps of the pattern and then students are asked to visualize the next steps, which involves both arithmetic, algebraic and geometric thinking.

 

Desmos in an online graphing calculator that allows for students to predict, Screen Shot 2018-12-11 at 2.52.41 PM

visualize and graph linear relationships and functions and relations.

 

 

So what does it mean to be proficient with visualizing?

As we begin to work with the new proficiency scale across BC, we need to consider what it means to be proficient with visualizing to explore mathematical concepts in relation to the grade level curricular content. As more teachers across the provinces the the scale, we will have examples of student proficiency that demonstrates initial, partial, complete and sophisticated understanding of the concepts and competencies involved.

For example, a grade six student at the end of the year would be considered proficient with visualizing geometric transformations if they were able to follow directions to mentally translate, rotate and reflect a 2D shape and show or describe the resulting orientation/position.

Screen Shot 2018-09-28 at 10.02.24 PM

Some questions to consider as you plan for learning opportunities to develop the competency of visualizing:

How is the core competency of communication developed through the process of visualization? What different ways can students show and explain what they are visualizing – using materials, pictures or words?

How do the competencies of estimating and visualizing complement each other to support reasoning and analyzing in mathematics? How can using visual referents support estimating?

How can we help students understand the purpose and usefulness of developing visualization skills and strategies? What examples can we share of scientists and inventors that used visualization to develop theories and ideas?

What opportunities are we creating for students to practice and use visualization skills and strategies across different mathematical content areas such as geometry, measurement, number, algebra and functions?

~Janice

 

References

Thinking Through and By Visualizing (NRICH)

The Power of Visualization in Math by Jeremiah Ruesch

Spatial Reasoning in the Early Years: Principles, Assertions, and Speculations by Brent Davis and the Spatial Reasoning Study Group, 2015

Taking Shape: Activities to Develop Geometric and Spatial Thinking by Joan Moss, Catherine D. Bruce, Tara Flynn and Zachary Hawes, 2016

 

September thinking together: mathematics curricular competencies

Posted on: September 28th, 2018 by jnovakowski

For the 2018-19 school year, the “thinking together” series of blog posts will focus on the curricular competencies in the mathematics curriculum.  The “thinking together” series is meant to support professional learning and provoke discussion and thinking. This month will provide an overview of the curricular competenecies and then each month we will zoom in and focus on one curricular competency and examine connections to K-12 curricular content, possible learning experiences and assessment.

KDU_knowdounderstand

The curricular competencies are the “do” part of the know-do-understand (KDU) model of learning from BC’s redesigned curriculum.

The curricular competencies are intended to reflect the discipline of mathematics and highlight the practices, processes and competencies of mathematicians such as justifying, estimating, visualizing and explaining

The curricular competencies are connected the the Core Competencies of Communication, Thinking  and Personal & Social. More information about the Core Competencies can be found HERE.

 

Screen Shot 2018-09-28 at 9.45.26 PMThe curricular competencies along with the curricular content comprise the legally mandated part of the curriculum, now called learning standards. This means these competencies are required to be taught, assessed and learning achievement for these competencies is communicated to students and parents.

Something unique about the mathematics curricular competencies is that they are essentially the same from K-12. K-5 competencies are exactly the same with some slight additions in grades 6-9 and then building on what was created in K-9 for the grades 10-12 courses. Because they are the same at each grade level, to be assessed at “grade level” they need to be connected to curricular content. For example, one of the curricular competencies is “estimate reasonably” – for Kindergarten that will mean with quantities to 10, for grade 4 that could mean for quantities to 10 000 or for the measurement of perimeter using standard units and for grade 8 estimating reasonably could be practiced when operating with fractions or considering best buys when learning about financial literacy.

The new classroom assessment framework developed by BC teachers and the Ministry of Education focuses on assessing curricular competencies and can be found HERE.  A document outlining criteria categories, criteria and sample applications specific to K-9 Mathematics can be found HERE. The new four-point proficiency scale provides language to support teachers and students as they engage in classroom assessment.

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As we are begin a new school year and are thinking about year plans and overviews we might consider the following questions:

  • What opportunities do students have to learn about what it means to be a mathematician and what mathematicians do?
  • What opportunities can be created over the school year for students to name, be aware of, practice, develop and reflect on the core and curricular competencies in mathematics?
  • How can we make the core competencies and curricular competencies in mathematics visible in our classrooms and schools?
  • As we are planning for instruction and assessment, how are we being intentional about weaving together both curricular competencies and content? What curricular content areas complement and are linking to specific curricular competencies?

~Janice