numeracy Archive - Mathematics and Science in SD#38 (Richmond)

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big mathematical ideas for grades 3-5 2019

Posted on: March 13th, 2019 by jnovakowski

This is the sixth year of this after school series that focuses on the big mathematical ideas encountered by teachers working with students in grades 3-5. This year this group met three times during term three.

Our first session was on January 17. Each teacher received the professional resource Number Sense Routines by Jessica Shumway.

The focus of our first session was on multiplicative thinking and computational fluency.

We began by working on a math problem together, from the book, and considered the different ways our students might engage with the mathematics.

And then looked at visual routines from the book that support multiplicative thinking through spatial structuring.

We also considered games that provide purposeful practice for developing computational fluency and reasoning around multiplication, such as the array-based game, How Close to 100? from Mindset Mathematics.

Our second session was on February 7 and after sharing the visual routines that we tried with our students, we discussed the big ideas around decimal numbers.

Our focus from the book was using number routines such as Today’s Number as well as Number Talks with fractions and decimal numbers. We also connected using visual supports such as 10×10 grids in games to practice decimal computation and develop understanding of decimal numbers in both fractional and place value-based ways.

Some games and a recording sheet for thinking about decimal numbers from the session can be downloaded here:

decimal_number_practice

Our third session was held on March 7 during which we focused on the big idea of area, connecting this concept to both multiplication and the visual routines we had learned earlier in the series (arrays, spatial structuring, decomposing into parts).

We also focused on the instructional routine of notice and wonder and how it can be used to have students make sense of a mathematical situation or problem as well as create an opportunity for students to ask questions that can lead into mathematical investigations.

Annie Fetter of the Math Forum has made many math teachers aware of Notice and Wonder over the years and an overview document is available:

Intro I Notice I Wonder NCTM

For this session, a new SD38 math instructional routine poster was created and it is available in both English and French:

notice wonder poster

notice wonder poster french

These posters are also all available on this blog, under the poster tab at the top!

Thank you to Grauer Elementary for the use of The Nest to host this series!

~Janice

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.

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.

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.

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.

Our 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.

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

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.

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.

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.

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.

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)

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:

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

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

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.

The 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

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.

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

Osmo 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?):

WODB_AR (movie file)

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

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.

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?

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

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.

October 25

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).

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.

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

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:

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.

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

One 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.

Thanks to the Diefenbaker team for hosting us!

~Janice

October thinking together: estimating reasonably

Posted on: October 31st, 2018 by jnovakowski

“Estimate reasonably” is one of the mathematical curricular competencies under Reasoning and Analyzing, the first strand of curricular competencies. The curricular competency of being able to estimate reasonably is a learning standard at every grade level from K-12. Because the curricular competencies in mathematics are not grade specific, they need to be connected to curricular content to be assessed and evaluated at grade level. For example, estimating reasonably:

at Kindergarten could be estimating within quantities to 10,
at grade 4 it could be computational estimation when adding and subtracting numbers to 10 000 or estimating the order of fractions along a number line using benchmarks
at grade 8 it could be estimating answers when calculating with fractions, estimating the surface area and volume of regular solids or estimating best buys when using coupons (financial literacy)

Curricular competencies to connect to many areas of curricular content but not all. When planning mathematical learning experiences, it is important to consider what competencies complement the content. For example, there are connections to estimating working with number concepts such as quantities, fractions and percentages as well as computational estimation, financial literacy and measurement.

Another consideration is that because this curricular competencies is the same essentially from K-12, it can be used as an access point for all students when planning for multi-age or cross-grade classes, developing IEPs and looking at class profiles.

What does it mean to be able to estimate reasonably?

As students begin their development of competency in estimation, they are comparing quantities as being more than or less than a known quantity. This further develops in using a referent for estimating such as if you know a handful of cubes is 10 cubes, you can use this information for estimating the total quantity of cubes in a jar. Likewise, a personal referent of knowing the size of your step that is about one metre long can help you to estimate distances. As students develop a strong sense of number, they are able to estimate within a reasonable range, knowing which numbers are too high and too low. As students become more competent with estimation and knowledgeable about quantity and other math concepts they are able to apply more abstract estimation strategies such as approximation and rounding.

How can we assess a student’s competence in estimating reasonably?

The Lower Mainland Mathematics Contacts network began to develop assessment tools to use with students to assess the curricular competencies. A draft of the estimating tool is here and teachers might find it a helpful starting place in thinking about how estimation develops along a continuum and the types of “I can” statements that can be used with students for self-assessment:

Estimating Ideas – LMMC DRAFT 2016

This assessment tool is still in draft form as we put this project on hold while the Ministry was developing a classroom assessment framework. General information about the classroom assessment framework, developed in collaboration with teachers, can be found HERE and the information specific to mathematics can be found HERE. The mathematics classroom assessment framework includes criteria categories and descriptors as well as examples from across grade levels. The Ministry is now using a four-point proficiency scale to provide descriptive feedback to where students are in their development.

Some resources to support competency development in estimation:

Andrew Stadel curates a website called Estimation 180 that is full of estimation tasks with a photograph as a starting point. Students are asked to consider what number would be too low and then which would be too high to develop their reasoning around what a reasonable range would be.

Many “three-act tasks” involve an element of element. Both Graham Fletcher and Dan Meyer have archived videos and examples of three-act tasks.

Steve Wyborney has developed a series of estimation tasks using photographs called Estimation Clipboard. You can download the slides and find more information about this instructional routine HERE.

For our BCAMT Reggio-Inspired Mathematics project, we have create a pedagogical content knowledge four-pager about estimating. You can download it here:

estimating-final

Two favourite picture books that focus on estimation, with a focus on using visual referents are Great Estimations and Greater Estimations by Bruce Goldstone.

Other picture books to connect to estimation:

How Many Seeds in a Pumpkin? by Margaret McNamara

Counting on Frank by Rod Clement

Betcha! by Stuart J. Murphy

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

Do students understand what it means to estimate, that there is reasoning involved?

How can we connect the curricular competencies of estimating and visualizing? Are students scanning quantities and using visual referents? How can we encourage students to explain their strategies and make what they are doing in their mind visible?

What opportunities can we create for students to make adjustments to their original estimates based on new information? Are they making meaning of the situation?

What opportunities are we creating for students to think about estimation across math content areas – number, quantity, measurement, financial literacy and other areas in context?

~Janice

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.

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.

The 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.

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

the new playground at Grauer: where’s the math?

Posted on: September 18th, 2018 by jnovakowski 1 Comment

Last year the families, staff and community fundraised for a new playground for Grauer Elementary. Grauer is a small school with only five, six or seven divisions (depending on the year) and it is hard work for a small school to raise $60 000! It was very exciting when the school reached their goal and is such a good example of an authentic numeracy experience for students to think about. In the BC curriculum, numeracy is defined as an application of mathematics to solve or interpret an issue or problem in context.

Last Saturday, I joined staff, parents and community members coming together to install the playground (self-installation with staff support from the playground company saves thousands of dollars). As Ms Partidge and I helped to read the specifications for the installation of one of the fire poles, we commented to a couple of parents around us how much mathematics was involved in the process.

I shared some of the photos from the installation day with the two grades 1 & 2 classes. All of these students had been to The Studio last year with me and had spent some times exploring the idea of “what is math?” so I framed this investigation as “where is the math?” I knew for some students this would create some dissonance as even young children can sometimes already have a very narrow view of what mathematics is and think that it is about counting, numbers and “plussing”. Part of this investigation was to disrupt this thinking. Of course counting, numbers and arithmetic operations are important content areas of mathematics, but they are not the only content. This investigation was one avenue to create meaning for learning mathematics, having students make connections to math beyond the walls of the classroom. The students came up with some initial ideas and we will continue to add to our thinking over the next couple of weeks.

The students were invited to design and create playgrounds and to consider where, when and how mathematics would be applied/used. One group of students followed the kit diagrams to create a Playmobil playground set – there was lots of math talk during that collaboration! Some students chose to draw and paint a playground from their imagination and some built playgrounds with blocks and loose parts, including a playground for animals.

After our first time together, I noticed the students were very interested in the photographs of adults using the levels and measuring tapes so I ordered some (not toy) tools to add to the construction area of The Studio. It was great to watch the students use these tools in authentic ways.

One of the classes had gone outside to look closely at the playground twice, creating detailed labelled diagrams or maps of the playground.

We extended this experience in The Studio by asking the students to create “math maps” indicating “where’s the math?” on recordings of their playground creations.

And what are are we assessing in terms of mathematics? These types of investigations and explorations lend themselves to informal formative assessment and gives us a sense of mathematical language the students have and where students are along a learning trajectory around different concepts and skills such as spatial reasoning, comparison of size and quantities and measuring. This type of assessment, that focuses on observing and listening to the students’ play and math talk is so important at this time of year and informs our instructional plans and focus for the fall.

When students engage in this type of learning through materials we make their learning visible through a sharing session at the end of our time together and capturing photographs, videos and students’ thinking so that we can revisit and reflect on the experiences, make connections to new learning experiences and consider questions for further investigation. The following are examples of documentation panels that we create to post in The Studio to help make our learning visible.

Grauer_playground1_Sept2018 copy

Grauer_playground2_Sept2018

I’m looking forward to seeing where the students take us next on this investigation.

~Janice

summer professional learning and reading 2018

Posted on: June 29th, 2018 by jnovakowski

Although summer is a “break” from the schedules and routines of teaching, it has always also been a time of learning for me. Whether it be taking course work or having the time to read deeply or attend professional learning events, I find the summer a great time to learn new things and both reflect on and rejuvenate my teaching practice. Of course, in order to really refresh, I do take some time away from professional thinking by reading novels, memoirs, travel guides and cookbooks! I try and learn new things and am currently enjoying learning about different types of weaving, dyeing using natural materials, using new art techniques and focusing on developing my knowledge around local plants All of these personal interests do tend to find their way into my professional work though as well!

One learning goal I have for myself is to become more familiar and fluent with using desmos. Desmos is an online graphing application (and available as an app as well) but has so many possibilities for supporting mathematical thinking for elementary and secondary students. The desmos website is full of examples and ideas for student projects as well as resources for teachers. I feel I just have a beginning understanding of what desmos has to offer so am looking forward to digging in and playing with it over the summer.

Professional Reading

My first summer professional reading stack of the summer!

Engaging Children: Igniting a Drive for Deeper Learning K-8 by Ellin Oliver Keene

Lifelong Kindergarten: Cultivating Creativity through Projects, Passion, Peers, and Play by Mitchel Resnick and Ken Robinson

Play Matters by Miguel Sicart

Arithmetic by Paul Lockhart

Give Me Five!: Five Coach-Teacher-Principal Collaborations that Promote Mathematical Success by Janice Bradley

Essential Assessment: Six Tenets for Bringing Hope, Efficacy, and Achievement to the Classroom (Deepen Teachers’ Understanding of Assessment to Meet Standards and Generate a Culture of Learning) by Cassandra Erkens and Tom Schimmer

Softening the Edges: Assessment Practices that Honor K-12 Teachers and Learners by Katie White

I have also ordered these two need mathematics book through the NCTM and the ATM.

An area of focus in our district will continue to be assessment. Continuous assessment that leads to responsive, intentional instructional choices is a practice that is woven throughout series I do around mathematics professional learning. Two books that I am going to revisit this summer as I begin to plan professional learning experiences for next year include:

Rethinking Letter Grades is a book by Canadian authors with local examples and I appreciate the “triangle” from this book that shares that in order to have authentic evidence of learning you need three types of assessment data – observations, conversations/interviews and products (which includes projects, creations, writing, drawing, diagrams, quizzes, tests). The Formative Five is a mathematics specific book focusing on five formative assessment practices.

New assessment reads for this summer include the following:

Katie White, author of Softening the Edges, will be a featured speaker at our Curriculum Implementation Day in Richmond next year. Essential Assessment was a book recommended by Angie Calleberg of the BC Ministry of Education as she said the Ministry used this book to inform assessment projects in the province. And although I do have some concerns about Hattie’s use of statistics and his meta analysis of meta analysis studies, I know his new book will come up in professional conversations around assessment so want to have a quick read through it.

Professional Learning Opportunities

For Richmond educators, professional learning opportunities are listed within the portal. Go to Learn 38 then to the Professional Learning tile to find both internal and external events.

For this year’s BCTF PSA Day in October, consider attending the Northwest Mathematics Conference in Whistler. Information about speakers, accommodation and registration is now available here:

Northwest Mathematics Conference website

Also in October, the Vancouver Reggio Association is hosting Tiziana Filippini, a pedagogista from Reggio Emilia, Italy. More information available here:

Vancouver Reggio Association – Tiziana Filippini – October 2018

A free professional learning event about coding for teachers is being hosted in Vancouver this summer, sponsored by the Government of Canada:

Teachers Learning Code – Vancouver – July 24-26 2018

Lots of districts in BC offer professional learning events at the end of the summer so check Twitter, Facebook, the BCTF site and district websites for more information.

For those of you interested in building your own knowledge of Indigenous perspective, culture and content, Talasay Tours offers some grant opportunities:

Talasay Tours – Authentic Cultural and Eco Experiences

And the Museum of Anthropology at UBC currently has an exhibit highlighting six cultures from across BC;

MOA – Culture at the Centre

Have a lovely summer – a time for adventures, rejuvenating and learning new things!

~Janice