Archive for November, 2014

thinking about factoring in grades 5 & 6

Posted on: November 27th, 2014 by jnovakowski 3 Comments

The students in Gillian Ewart’s grades 5 & 6 class have been beginning to learn about factors and multiples through creating arrays. Last week we decided to play around with the concept of “halving and doubling” as a strategy and how it could help us think about factors. I asked the students to created an array for 8 x 6. This task brought up language around columns and rows and what 8 x 6 would look like.


We then took the students through modelling halving and doubling with their arrays. Halving the numbers of rows and then sliding one half of the rows up to double the amount in each new row.



The big idea here is that the product (48) stays the same and that there is a relationship between halving and doubling. The students began to anticipate what their arrays were going to look like, realizing they were going to need to be creative in order to create their arrays after halving and doubling their 2 x 24 arrays!


As the students built and photographed their arrays, we recorded the corresponding equations on the whiteboard.


We also looked at all the factors we found for 48, beginning to look at factorization and “factor trees”.


As students were halving and doubling, they documented each stage by taking a photograph with the iPad and then labelled their photos using the Skitch app. We asked the students to focus on communicating what they had learned about factors by using arrays. The students then either used PicCollage or ShowMe on the iPads to compile their photographs and share what they had learned.


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Here are some examples of the PicCollages the students created:

40C00C3A-3B9ACA00-1-PicCollage BD2FA242-3B9ACA00-1-PicCollage 88BC7CD3-3B9ACA00-1-PicCollage 036C951C-3B9ACA00-1-PicCollage F3b600B1-3B9ACA00-1-PicCollage 7D416969-3B9ACA00-1-PicCollage

As students completed their documentation as a way to share their learning, they were asked to choose a new multiplication equation and play around with the idea of halving and doubling their arrays. This is something that the class was going to continue investigating after I left. We could have begun our investigation this way, with students creating different arrays, halving and doubling and seeing if they could generalize what might happen. For this context, we decided that a guided approach to start would provide the students with the language and understanding they needed to be successful when they investigated their own arrays.

This was my last scheduled visit to McNeely and our goal was to introduce a variety of iPad apps to the students so that they would have different ways to communicate their mathematical thinking and learning. I’m looking forward to hearing from the classroom teacher and students as to how this journey continues for them.


introducing ten frames

Posted on: November 25th, 2014 by jnovakowski

I visited Susan Carrusca’s grades 1 and 2 class at Byng Elementary to introduce some different ways to use ten frames to support visualization leading to working with mental mathematics strategies. Byng’s school goal is around developing computational fluency and greater student engagement in mathematics

ten frames


Many of the tasks we did together were based upon ideas from a Math Solutions book – It Makes Sense! Using Ten Frames to Build Number Sense.


We came together on the carpet area and I used Trevor Calkin’s large Power of Ten cards to play Flash It with the students. I held up a ten frame card for a second and asked the students to call out what number they saw represented. We talked about not trying to count but to take a picture with your brain. We discussed what numbers were easier and why (9 was easy for them because the students said they saw one space empty so they knew one less than 10 was 9) and which ones were more difficult – 7 and 8. Using a large magnetic ten frame on the whiteboard I showed them the importance of “building on 5” and seeing 7 and 8 as five and something (5 and 2, 5 and 3). We played Flash It again, and the students were more confident (and louder) at recognizing the numbers.

The students then moved back to their desks where they each had a ten frame mat and a basket of loose parts/counters. We played Show It, where I held up a ten frame  card long enough for all students to have a look and then the students showed that number by building it on their ten frames.


Using the same ten frame mats and loose parts, the students played Roll it, Build It by rolling a regular dice. Simply, they rolled the dice and then built that number not their ten frames. After a few rolls we extended the task by asking the students to visualize using the ten frame “how many more to make 10?” for each roll and build.


We came back to the carpet to work on combining numbers and the strategy of Making 10. Using 8 + 5 below as an example, I build 8 in one ten frame and 5 in the other, writing the equation below. I asked the students how we could use the ten frames to help us “make 10”. A student suggested counting on and another suggesting combining the two 5s across the top (using doubles, but also making a 10!) and then adding the three. After recording those ideas, I asked the students then to connect to what we just did with the ten frames. If we had an 8, how many more would we need to make 10? They all called out 2 and I asked them where they could take a 2 from. The students began to see they could “decompose” the 5 and take a 2 from it to make the 8 into a 10, leaving us with 10 and 3 to make 13.


That may seem like a long way around to get the answer to 8+5. I should have mentioned that most of the students knew the answer was 13 before we started our discussion. Some counted on from 8 with their fingers or in their heads and others just “knew” the answer. Knowing basic facts is essential but knowing mental mathematics strategies is even more essential and powerful. By developing strong number sense and a repertoire of thinking strategies, students will be able to figure out questions like 28 + 5 and then 48 +35 in their heads because they can apply strategies like “making 10” as we did here with 8 + 5. We want students to understand the mathematics they are doing and to become fluent and flexible in their thinking.

As a final practice task, the students played Race to 20 using a double ten frame mat, loose parts and a regular dice. For this game, they shared the dice with a partner, taking turns rolling. Each student build the number he or she rolled and built it on the ten frame. For this game, they didn’t clear their boards after each roll, their total accumulated with the goal of reaching 20. As students played, we asked them to visualize what their double ten frame mats would like, before they added the loose parts. They continued playing and the classroom teacher and I circulated amongst students, asking them to explain how many more they needed to roll to make 20 and how they knew that. The students that were able to record their equation string did so on their game mat. This time, we played that you didn’t have to roll the “exact” number to reach 20, going over was fine.

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Throughout our afternoon together, we kept coming back to the idea of using the ten frame as a tool to help us think and to visualize.

A sweet end to the afternoon was having one of the young girls ask me quietly as we were tidying up the materials if she could take her ten frame mat home so she could play math at home. There was a lot of engagement and big thinking about mathematics all afternoon and it was great to hear that this student wanted to continue this “play” at home.


circular patterns inside and outside

Posted on: November 24th, 2014 by jnovakowski

Since my last visit to Marissa Kishi’s kindergarten class at Whiteside, the class has been continuing to learn about repeating patterns and Marissa has been reading parts of one of the books from the kit – Spotty, Stripy, Swirly: What are Patterns?


The class had read up to the page about patterns in nature and since it was such a sunny day outside, Marissa thought it would be great to go outside. I shared the page below with the students again to focus them on the circular nature of the patterns and have them think about what they might find outside.


The students started collecting treasures – beautiful autumnal leaves, pinecones and twigs.


After a loop around the school grounds, the students gathered together in a concrete covered play area and we asked them to see how they could use the materials to create patterns. While we might have hoped that students might have suggested this themselves or just began naturally creating with the materials, that didn’t happen so we prompted them. To me there is a difference in directing the students to “make a pattern” and to inviting them to investigate how they could use the materials to create patterns. Subtle but significant difference in that the inquiry and learning is turned over to the children in a way that engages them at a different level.

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What the classroom teacher and I noticed is that the students created linear, repeating patterns with 2 elements and almost all of their patterns were of the AB structure. They were “stuck” and this is what we had noticed in the classroom as well. Part of the constraints outside were the limited materials they had to work with. I noticed that two of the students each had a small bough of pine needles that had fallen off the trees during the last storm. The students hadn’t included these in their patterns as they only had one of the item. I asked if they put the boughs in the middle of a pattern, would it be possible to create a pattern around it, trying to connect back to the images we looked at in the book before we went outside. There was only interest in this from a few students. I was more demonstrative than usual about what these students were doing, hoping to draw some others students into the circular pattern creation. It worked.



The students rallied together, with “organizers” emerging and others happily collecting materials and adding to the patterns that were being created. A few students were quite hesitant to give up their special treasures so watched and protected their leaves and pinecones. The students created a very large mandala until the wind picked up and started to lift their leaves away. Time to go inside.

We laid out some materials from the Reggio-inspired patterning kit, trying to keep the focus on other ways to look at patterns. Students enjoyed the spiral mats, continuing with linear patterns but not in a “straight line” across the table.




Other students chose some “loose parts” to create circular patterns like we had worked together on outside. I loved watching how some students were very particular about symmetry, balance and pattern within their designs.

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By providing the students with some new patterning experiences, the intention is that this will help them to build a deeper understanding of what a pattern is – that there is some order, regularity, generalizability. I am looking forward to visiting this class in December to see what they are thinking about patterns at that time.


introducing playful storytelling at Ferris

Posted on: November 21st, 2014 by jnovakowski

The three kindergarten classes (one is a K&1 class) and their teachers are participating in our Quality Teaching and Learning (QTL project this year. We are looking at how playful storytelling experience support oral language development and understanding of story. Using natural materials and animals and stories from local, place-based contexts we are exploring how the First Peoples Principles of Learning can inspire our teaching practices.

For this project the principles we are focusing on are connection to place, the power of story and awareness of self-identity. The big idea is that we all have stories to tell.

The students were introduced to the materials in the “starter kit” that will stay at the school. We passed around the animals and read a simple story, Good Morning World.


The students were then able to create a place/setting for their story and choose some animals to help tell their stories.

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The following is the documentation panel that was created with student comments and reflections from the three classes responding to the question What is a story?

What is a story_ doc panel





Reggio-inspired mathematics: geometry kit

Posted on: November 20th, 2014 by jnovakowski

The fourth of our Reggio-inspired mathematics kits was enjoyed by the Kindergarten and Grade 1 students in Lauren MacLean’s classroom at Blair Elementary on Tuesday afternoon.

The materials arrive in a bin, including cork mats and baskets. This kit includes a variety of two and three-dimensional geometric shapes and is intended to be supplemented with materials from the classroom such as building blocks, pattern blocks, etc.


The materials were spread out over four tables – shape sticks (inspired by a blog post found here), 3D wood shapes, shape puzzles and pattern blocks. The picture books from the kit were placed on the carpet for students to have a look at.

As the students engaged with the materials, they were prompted with a question that has a curricular focus at these grade levels – How can we combine shapes to make new shapes?





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We also introduced Osmo for the iPad and the students used the Tangram app. Three Osmo devices are now available through loan through our District Resource Centre (DRC) and I have two that I can share with classrooms.

One of the things I like most about Osmo is that it fosters collaboration, problem-solving and creative & critical thinking. I love seeing a small group of children huddled around the Osmo/iPad combo, trying to figure something out together as Lauren’s little ones did, solving tangram puzzles.



One of the great new features of the Words app for Osmo is that you can create your own photo albums so the students in this class helped me take photographs of all sorts of shapes which we will create a photo album with to play Words with.

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The students came together and we shared how we combined different shapes together to make new shapes or things. We talked a bit about the materials and how math materials help us learn about shapes but that if we looked around we could see shapes everywhere in our world. The students were so curious and I could see their eyes wandering all over the classroom, noticing and pointing out shapes that they could see, leading to our next inquiry – What shapes can we see and how are shapes important in our world?


thinking about equivalence in grades 5 & 6

Posted on: November 20th, 2014 by jnovakowski

Last week during my visit to Gillian Ewart’s grades 5 and 6 class, we played around with the big idea of equivalence. The focus of our time together was coming to an understanding that the = symbol is a sign of equivalence or balance.

I began by reading the first few pages of the picture book One is a Snail, Ten is a Crab, pausing a few times to check in with students to see if they were “getting it”.



I wrote the equation 9 = 2D + 1 on the whiteboard, intentionally choosing the order in which I presented the equation. I asked students to talk to a partner about what I had written. They shared their thinking and they realized the D was for dog (D is 4 legs) and that the +1 was the “constant” of the snail, as expressed in the book. I asked if there was another way I could have made 9 and they shared that a S + 1 would also work (S is the eight legs of a spider).

As a whole class we played around a bit with the idea of the balance scale and balancing both sides of the equation. So I added another dog to the right side and asked what I need to do to the left side to keep the equation balanced….D + 9 = 3D +1.

I asked the students to create an equation and add different animals or amounts to keep it balanced. A student asked, “Do we have to write the equations like that? (pointing to the whiteboard) Backwards?” which led to a great discussion about equations and trying to really get at what the = sign means. Some of the students continued to refer to “5” in the example below as “the answer” which as teachers, gives us great information about the myth-busting we can work on together.



The students engaged in the balancing equations work in a range of ways. Some students got very creative and complex with their equations on their whiteboards, others modelled their equations with materials, some students were able to think about the big idea using less complex equations and with the support of an adult and others used patterns in their equations to build and extend.

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As students created equations, they took photos with the iPads to document their experiences. I did a quick demonstration of the PicCollage app and asked students to combine some of their photographs with a statement of learning for our time together. This is a great assessment check-in as a teacher and helps students to focus on big ideas in mathematics and providing evidence of their learning.




Reggio-inspired mathematics: measurement kit

Posted on: November 18th, 2014 by jnovakowski

On Thursday morning I visited Louesa Byrne’s Kindergarten classroom at Thompson Elementary to deliver our next Reggio-inspired mathematics kit which focuses on measurement concepts.


The kit contains a variety of materials to investigate different types of measurement – linear measurement, passage of time, volume/capacity. Additional materials and tools from the classroom that could supplement this kit are pan balances, stopwatches and various “non-standard” materials to measure with.

The first class to use the materials in this kit is Louesa Byrne’s Kindergarten class at Thompson Elementary.


Louesa was one of the teachers involved in our Reggio-Inspired Mathematics Project last year and her area of inquiry this year is the “what next?” piece…being responsive to what she is noticing while students are working with the materials. She is also curious about what inquiry projects might emerge in her classroom that may involve mathematics.

On Thursday, we began our time together by having a group discussion prompted by the questions, “What is measurement? What can we measure?” The students had all sorts of ideas – you can measure how tall you are which led to comments about all sorts of different things you could measure, such as cars. This led to my wondering aloud whether there was anything else you could be measuring while you were driving in a car. I was thinking about distance travelled but a little girl commented on speed. She was able to elaborate with explanations of how 60 was too fast, unless you were on a highway!

We talked about comparing to measure and I shared some pages from the classic book Actual Size by Steve Jenkins. I called a few students up to stand by the book to compare the size of a child’s eye to the ginormous squid eye, of our teeth to a great white shark’s teeth and finally, the gorilla’s hand, always a favourite! The gorilla’s hand is always such a good example to discuss the concept of baseline. We noticed that some of Louesa’s students have yet to develop the idea of conservation of size – that an object’s size doesn’t change because its position is changed. This will be something that Louesa can respond to as she considers math provocations for her students.

The students worked in pairs using a toy snake. The students were asked to find something longer and something shorter than their snake.

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As students were satisfied with their snake measuring, they were invited to explore some the measurement materials displayed on small table in the classroom.


This student wondered how long the sand would take to move to the bottom.


Many of the students commented they had seen similar dolls before.


This student wondered how many of the smaller cups of water would fit in the biggest cup.


When measuring, there are many opportunities to develop specific mathematical vocabulary and we overheard students comparing and discussing their findings with each other.


playful inquiry with numbers

Posted on: November 13th, 2014 by jnovakowski 1 Comment

I spent a delightful morning in Stephanie Merrick’s Kindergarten class at Hamilton Elementary on Wednesday. This was my first time meeting these students so we began with a quick little introduction to subitizing – I flipped over dot cards (dots arranged in various patterns) and asked them to take a brain picture and without counting, tell me how many dots there were.

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On the second round, we paused and I had students describe how they “saw” the dot patterns. For example, for this quantity of dots, students shared that they saw this as 4 and 1, 2 and 3 as well as 2 and 2 and 1.

To follow up on the concepts of subitizing and decomposition, one of the provocations presented to the students was to find all the ways to make 5. Students were provided with a numeral 5 as a reminder and large gems and cork mats to work on. Presenting the task as a question such as “Can you find all the ways to make 5?” promotes investigation and playful inquiry.

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At another table, a five frame “game” of building and changing was presented. Students rolled a die and then filled the five-frame up with that quantity (we problem-solved as a group what we would do for 6). The student then rolled the die again and either builded on their existing quantity (adding) or removed some gems (subtracting). This process can involve visualizing, counting on, subtilizing, decomposing – all important early numeracy concepts.


As time went on, I noticed students adapting the task to make it more suited to their needs. Some students worked as “teams” and combined their five frames and rolled two dice. Some students cleared their five frames after each roll, working on their counting and representing quantity each time rather than building and changing.


At another table, we simply put out a basket of dominoes and a tray of numeral cards. I gave a few prompts as to what students might do with the dominoes (matching, making a train, etc), The first group of students to settle in at this table were content to build.


Stephanie, the classroom teacher, sat down with the students at one point and prompted them to match the dot patterns on the dominoes, realizing that many of the children may not have played with dominoes before.


This led to some students doing their own individual matching of the patterns and connecting the numerals to the dot patterns.

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The last table we presented a variety of materials – pebbles, gems, open ten frames, dice, ten frame cards and wooden numerals. At my first pass by, a little girl wanted to show me her “Counting Fun House” and explained that she made a house and then rolled the die and that was the number of objects she put in the middle of the house. When I returned to the table a few minutes later, she mentioned the house was getting full and she needed to move the walls to fit more things in.


The students were not provided with any direct prompts for this table and it was fascinating to see the different ways the students used the materials.

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The students played with ordering the numerals, representing quantities, using ten frames to match quantities, counting, combining and building on five.

In the next few weeks, Stephanie will be documenting how her students have experienced the materials and I hope to share some of those glimpses into her classroom here!


primary teachers study group: session one

Posted on: November 10th, 2014 by jnovakowski

Because of the difficult start to our school year, I have been holding off on starting up the after school professional learning series that we host in the district, but our primary teachers study group seemed ready to get going so we had our first session of the year last week. As voted on by the group at the end of the year, we are continuing our professional inquiry into creative thinking, adding in critical thinking as the year continues and the work in that competency develops.

We had a look at how our Ministry of Education is defining the competency of creative thinking and discussing what the three significant facets are – novelty and value, generating ideas and developing ideas. We shared what we used to think about creativity and what we think now – many of us thought that creativity was directly related to artistic ability or creation of a product and now we realize that creative thinking is much broader and crosses curricular lines.


Playful inquiry with materials and ideas is a way to create opportunities for creative thinking. We shared what kind of environment we might need in order for students to be creative, take risks and be innovative. How can we nurture creative thinking in the classroom?


The notion of being creative with ideas, thoughts and words is something that some teachers may explore. Big ideas around story and use of language are important in our classrooms. How might we inspire more creative thinking in students’ storytelling or writing?

Teachers in a study group are able to purchase books at a study group rate. Some years we have used a teacher resource book but for the past few years, we have used picture books to inspire our work. This year we are starting with three titles, each meant to help us focus the facets of the Creative Thinking Competency. The picture books were introduced as starting points for classroom conversations and experiences around the facets of creative thinking – novelty and value, generating ideas and developing ideas.

Here are some ideas that we brainstormed together:

mix it upMix It Up!

by Herve Tullet

Invite students to investigate colour mixing using liquid watercolours, cups, jars or vases of water and eyedroppers.

Add colour to students’ investigations of the properties of matter.

As part of the Mind Up program – watching colours in water and discussing how feelings are connected to colours.

Play with mixing warm and cool colours.

Paint to music, using colours to represent.

Mixing and naming colours, using circles to represent the fractional parts of colours used and mixed.

if you hold a seedIf You Hold a Seed

by Elly MacKay

Introduce the idea of metaphor – that a seed could represent an idea.

This book highlights the notion that things take time to develop and patience and nurturing is needed. An interesting question to explore might be: What do ideas need?

The idea of vision or thinking with the end in mind is also a theme in this book.

what do you do with an ideaWhat to you do with an idea?

by Kobi Yamada

Brainstorm and discuss people’s ideas that have been transformative.

Have students share ideas and inventions that have changed the world or made someone’s life better.

Our next session is in January where will we be sharing some of the ways we have used the books in our classrooms.

Thank you to the Diefenbaker team for hosting us for our first session!



Reggio-inspired mathematics: number kit

Posted on: November 10th, 2014 by jnovakowski 15 Comments

On Wednesday morning I will be visiting a Kindergarten class at Hamilton Elementary to deliver and introduce the Reggio-inspired mathematics kit looking at number. The kit arrives looking like this…are you wondering what’s inside?


Here is an example of one way to to present the materials:


Ideally, a spot in the classroom could be found so that the materials could be left for students to access when they need or want to. Materials can be pulled to spots in the classroom to set up specific provocations during a math learning time or during a general “centre” time.

In this kit you will find a variety of materials for young students to develop one-to-one correspondence, count, subitize, compose and decompose quantities and represent numbers in different ways.


Easels are provide to present inquiry questions, books or images. The image above shows a focus question to inspire students to investigate different ways of making seven using large gems. Mats of cork or felt provide a boundary for students as well as cushion the sound of the materials on a hard surface.


Wood numerals and printed numerals on cardstock are provided for students to label their representations. Some students enjoy ordering the numerals as well.


A variety of visual tools such as dot patterns on cards, dice and dominoes are in the kit in order to support the development of subitizing (the instant recognition of a quantity of objects/dots). Ten frames made of popsicle sticks are also included. A visual tutorial of how to make these ten frames can be found HERE.


There are picture books and “loose parts” to inspire playful inquiry with numbers.

Part of the professional inquiry that our early primary teachers are entering into this year is considering how Reggio-inspired practices might shift how we view the teaching and learning of mathematics. We will be trying some different approaches with the materials in classrooms and reflect together on how these approaches are supporting student learning and engagement.

More to come…