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

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.

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.

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

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

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

playful_inquiry_charcoal_2018

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.

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

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.

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.

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.

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.

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:

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

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.

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

The 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 composition 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

Fawn 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,

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.

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

November thinking together: develop mental math strategies

Posted on: December 2nd, 2018 by jnovakowski

Develop, demonstrate and apply mental math strategies

is the focus of one of the Reasoning and Analyzing curricular competencies from grades K-9. For K-5 the focus is on developing mental math strategies as a means to developing fluent and flexible thinking with numbers. In grades 6-9 the focus is on demonstrating and applying these whole number strategies to new number contexts.

As is the case with all the mathematics curricular competencies, the learning standard is the same for K-5 and then continues for grades 6-9. Grades K-5 focus on the development of mental math strategies while grades 6-9 focus on the application of mental math strategies. The grade level-ness is enacted when the curricular content and curricular competency are connected.

So for example, in grades 6-9, the competency using mental math strategies with whole numbers is applied to decimals in grade 6, integers in grade 7, fractions in grade 8 and rational numbers in grade 9 and it is this intersection of curricular content and competency that is assessed.

The suggested mental math strategies are listed in the elaborations for the curricular content for each grade. Elaborations are suggestions and support for instructional decision-making and are not meant to be used as a requirement or an assessment checklist. Mental math strategies are strategies that are intended for students to be able to do mentally. Computational fluency involves  flexible strategy use – both mentally and recorded with paper/pencil, whiteboards, etc. The strategies are transferable to working with larger numbers or to different types of numbers such as fractions and integers. Students may be introduced to the strategies by their peers during a number talk or during an instructional task. When strategies are introduced, specific mathematical language and visual scaffolds such as ten frames or number lines are often used. Some strategies may need to be practiced in different ways before students are able to use them mentally in flexible ways, and choose strategies that make sense for the numbers they are working with. As an example, the following are the computational strategies suggested in the content elaborations for grade 2:

In our 2007  curriculum (WNCP) there was a clear definition of what Mental Math is (in the 2007 iteration of our curriculum, it was named as a mathematical process). “Mental mathematics is a combination of cognitive strategies that enhance flexible thinking and number sense. Mental mathematics enables students to determine answers without paper and pencil. It improves computational fluency by developing efficiency, accuracy, and flexibility.”

Goals of developing fluency with mental mathematics include:

• developing confidence in doing mathematics
• being liberated from calculator dependence
• becoming more flexible thinkers
• be more able to use multiple approaches when problem solving

(Rubenstein, 2001)

Strategies develop over time and complement each other. Examples of mental math strategies drawn from the elaborations in our BC curriculum framework include:

• counting on
• making ten
• decomposing (to make tens/hundreds, by place value)
• double and related doubles (doubles plus one, etc)
• bridging over tens (transferable to hundreds, thousands etc)
• compensating
• adding to find the difference
• commutative, associative and distributive principles
• annexing zeroes
• halving and doubling

Different visual tools can be used to support students’ development of mental math strategies such as ten frames, hundred grids (numbered and blank) and open number lines.

Teachers in Richmond have been developing their own understanding of mental math strategies through Number Talks over the last several years and I believe it is the most used instructional routine in our K-7 classrooms.

The following is a record of some of the number talk experience in Richmond classrooms.

SD38 Number Talks panel 2016

Carrie Bourne and I have started creating a math video series for Richmond educators that our available on our district portal. The series is called Doing the Math Together and the videos are intended support teachers with their understanding of mental math strategies and how they might record students’ strategies during number talks. For example, there are videos in both English and French on the importance of the “making ten” strategy.

They are located in the district portal on the Mathematics page, under the blue tile titled Doing the Math Together Video Series.

Professional resources to support the development of mental math strategies through the development of number talks include:

So what does it mean to be proficient with mental mathematics?

As we begin to work with the new proficiency scale across BC, we need to consider what it means to be proficient with developing, demonstrating or applying mental mathematics 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 2 student at the end of the year would be considered proficient in adding two-digit numbers mentally if they were able to be efficient, accurate and flexible when using two or more different mental math strategies such as decomposing or compensating.

Some questions to consider as you plan for learning opportunities to develop the competency of using mental math strategies:

What strategies or knowledge do students already have about mental mathematics? What opportunities do students have to show and share what they know?

How can we encourage students to be metacognitive when using mental mathematics rather than just applying procedures? How can we develop the concept of efficiency and support students in choosing strategies that are a good fit for the numbers provided?

How can we help students understand the purpose and usefulness of developing mental math strategies?

What opportunities are we creating for students to use mental mathematics across other disciplines such as science or ADST?

~Janice

References