Archive for April, 2015

more measuring in kindergarten

Posted on: April 25th, 2015 by jnovakowski

One of the things that is fascinating about using open-ended provocations in mathematics is that every experience with the materials is so different. Materials are chosen intentionally and often set out together to suggest an investigation but where the students take things makes it their own and often goes much deeper with the mathematics than what we may have intended.

Earlier this month I spent part of a morning in Stephanie Merrick’s kindergarten classroom at Hamilton elementary and the students had been using the materials from the Reggio-inspired measurement kit. The students in the class listen carefully to each other and are experienced with engaging with materials both independently and collaboratively.

I did a short mini-lesson on comparing linear measurements reading a book called Big and Small that compares different sizes of animals. We discussed how the term “big” is too general and that we need to use more specific language like longer, taller and shorter when comparing, I then modelled this using a set of matryoshka dolls, ordering them and using the comparative language. The students were then invited to use the materials and investigate measuring and how to compare measurements.

The provocations placed out on the tables…

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And the students began measuring and talking about their measurements. We also placed some tubs of materials on the carpet. One student sat down not the carpet and started measuring his leg with cubes which then inspired others to measure parts of the themselves.







This pair of students lay down the panda matryoshka dolls and measured their lengths with gems and then recorded the measures.


As the girls were undoing the pandas to put them back together as a set, they began filling them with the gems they had been measuring with and wondered about the capacity of each doll.


During this visit I had a Queen’s student working with mean she interviewed Stephanie and captured some of her thoughts about teaching mathematics through Reggio-inspired practices. Specifically, Stephanie was asked how she thought student learning was affected.

  • “Having everything hands on and play-based lends itself to extension. It lends itself well to students finding their own extensions”
  • The practices and materials help students, “learn where they are at.  If students are ready to learn more, they will explore it naturally.”
  • The teacher noticed that it “takes away from my own micromanaging as they find where they will go next independently.”
  • “It is engaging.”
  • “It is easy to say to students ‘can you show me that in a different way with a different material?’” and the practices and materials provide these prompts.”

The students were engaged with the materials for almost an hour. During that time both Stephanie and I were able to sit alongside every child in the class and capture a glimpse of their understanding about measurement. The current prescribed learning outcome for measurement in kindergarten here in BC is: use direct comparison to compare two objects based on a single attribute such as length (height). Every student in the class was able to compare two objects (ie. two ribbons, two of the matryoshka dolls, their legs or arms, etc) and use the specific math language of taller. longer or shorter. Many students demonstrated understanding well beyond this outcome and measured using non-standard and standard units and were able to explain the differences in lengths or heights of the objects they measured using units. For example, one of the matryoshka dolls was eight cubes tall and another was four and a student said that the taller was was four cubes taller. As Stephanie suggested in her comments above, the provocations and materials provided the opportunity for students to extend their own learning.


math in our community

Posted on: April 25th, 2015 by jnovakowski

Many of the classes at Byng Elementary are participating in a place-based mathematics inquiry project, support by the Aboriginal Enhancement Schools Network. Big ideas we have been looking at include seeing where math lives outside of the classroom including the school grounds, the neighbourhood, the community and the river. As we take students out for math walks, we begin with the strategy of noticing and naming where we notice math. The grade 1 class was very focused on noticing and naming shapes! For the older classes, we also focused on trying to help the students connect to place and when we were down by the river, I shared the story of this place and how it was shared fishing territory for the Coast Salish nations for thousands of years. Big ideas of time and place as well as how mathematics is used in the community will continue to be investigated.

Looking for math in the neighbourhood…


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Looking for math in Steveston village…




The grades 5 & 6 class had a chance to talk to three RCMP officers about they use math in their jobs.


Looking for math at the harbour…




As each class walked through Steveston, the students used the school’s iPads to take photographs. When we returned to class they had to edit down their collection of photographs and then talked about what mathematical connections they were making to their photographs.



We will be continuing to think about these photographs and think about what mathematical investigations might be inspired by them.


Montessori mathematics

Posted on: April 25th, 2015 by jnovakowski

Earlier this month I met with two intermediate Montessori teachers from each of our three elementary schools with Montessori programs. We discussed issues related to the redesigned mathematics as they pertain to teaching with the Montessori philosophy and materials.

Coincidentally, later that week the teachers at McKinney were hosting a Montessori math night for parents from all three schools. Fortunately, I was able to attend. Two of the teachers provided an overview to parents and then we were able to see all the specialized Montessori mathematics materials. They were presented on tables filling up the school gym and students from the school were at each table doing demos and explaining how to use the materials and what math concepts were being developed.





Here is a short Animoto video with some of the students explaining the materials. LINK HERE.


a kindergarten building project

Posted on: April 5th, 2015 by jnovakowski

Louesa Byrne’s Kindergarten class at Thompson Elementary generally spends a good part of their afternoon learning time dedicated to project work. Students investigate their own inquiries, often working in small groups. Last Wednesday afternoon when I visited the class, the students were going to engage in whole-class project time, pursuing related projects together. Louesa has had the Reggio-inspired geometry kit and the students have enjoyed building with the materials, but there are small amounts of materials included and this is not conducive to whole class construction.

Louesa noticed the students interest in the geometric materials and had also been noticing how some students were “stuck” when working with the construction materials in the class in that they usually built the same kinds of things. She also noted that some of the students did not choose the construction materials at all. As a way to inspire her students to construct and build with materials as well as to consider the learning outcome for geometry in kindergarten (attributes of three-dimenstional objects) she developed the building project.

The class had also just begun a study of community and some drawings of buildings and places that were important to them were posted in the classroom so the student had already begun some thinking around buildings.

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Louesa began by showing the students colourful images of buildings from around the world. These images were also presented alongside construction materials around the classroom. Louesa asked the students to consider which materials would be good choices for different structures and in making different shapes. She provided each pair or small group of students with a large piece of paper on which to construct their buildings, to provide a space and boundary for them.

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The students then chose what materials they wanted to work with and spread out around the classroom. Many had their own ideas about what they would build while others looked to the images, some for inspiration and others tried to re-create the buildings.

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As the students built, they were noticing each others’ constructions. As is typical for this age, I overheard one student exclaim, “Their’s is higher!” and immediately turned back to his tower to find a way to make his taller. This was an opportunity to for me to provoke his thinking about measurement but I didn’t, I sensed that he was very in the moment with the competitive aspect of his building and didn’t want to pull him away from that.

Louesa paused the students and passed out pieces of white paper for observational drawings of their buildings. I found it fascinating that for some of the students they went from looking at a 2D images, constructing a 3D building and then back to 2D representation. Lots of interesting things happening in their brains! One student drew a square on his paper and then looked at me and said he didn’t know how to draw the next part of his building because “it went up” and he didn’t know how to make it come up off the paper. I asked, “How else could you look at it? What shapes can you see?” and I slightly moved my head to the side to model looking at his building from a different perspective. With this, he was able to record a side-view of his building. Another student, who had created the Great Wall of China recorded a birds eye view of her construction.

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I noticed the students using some rich mathematical language and discourse even though they didn’t approach this project as a “math lesson”. There was talk of curved and straight edges, what 2D shapes they noticed in 3D shapes, how a cube shape created with straws and connectors was “like a dice” and how certain 3D objects stacked on others while others didn’t. There was also a lot of visual-spatial awareness and problem solving as well as elements of comparative measurement.

As it got close to 3pm, Louesa let the students know they had a few more minutes to complete their buildings and drawings. She then led them on a walk around the classroom so they could have a look at each others’ buildings.


We then gathered on the carpet to have a debrief of the project time. Louesa and I shared some of the things we noticed such as teamwork, learning as you were going and problem-solving as well as more specific things to this project such as thinking in 2D and 3D and drawing from different perspectives. This was such an engaging learning experience for the students and not once did I feel a sense of being rushed or in having to finish or complete a particular task. The learning and joy was in the process of the experience.


where does mathematics live outside?

Posted on: April 3rd, 2015 by jnovakowski

Byng Elementary has been involved in a place-based mathematics project and we are beginning our second year of exploring ways that students can make connections between mathematics, place, story and culture. The school has been part of a project through the Aboriginal Enhancement Schools Network (AESN). On Wednesday morning, I visited the two kindergarten classes and we introduced some big ideas.

We began with a sharing circle focused on the question: What is mathematics?

Most of the students commented that it was numbers, counting, adding and subtracting, putting together and one Kindergarten student even said math was homework! With some coaxing I was able to get the students to acknowledge that shapes were part of math too and they were able to name some shapes. One student said mathematics was thinking and learning…that made my heart sing a little! I needed to jump in and do a little discussion about mathematics being more than working with numbers and I listed several things that were “mathy” trying to help the students make connections.

With a beginning understanding that mathematics is more than numbers, we took a walk outside and huddled together near some large trees. I asked students to look around like mathematicians and notice where they see mathematics or make a connection to mathematics. This was not easy for these students. I think for these classes, there was some novelty about being outside even though I had carefully prefaced our trip outside as a “thinking and learning time” there was still a lot of distraction, jumping on rocks, climbing trees, digging worm holes, wanting to run, etc. You know, typical five year old stuff! It reminded me how important it is for students to be outside regularly and to just see the outdoor as an extension of their classrooms.

One class was very focused on looking for numbers and commented that the trunk of a tree looked like a 1 and that one of the rocks, if you turned sideways, looked like a 1.


The students displayed some curiosity about the two large evergreen trees and how long they had been there. They noticed how tall they were and one student wondered how you could measure them. In trying to connect to the age of the trees,  I shared the story of this place and tried to have the students imagine what this place might have looked like before the school and playground were there, before there were roads and before there were houses and stores. This was hard for these students. One student thought there would just be dirt. I shared the story of the place near Byng, the river and what is now called Garry Point. I explained that for thousands of years a community of people called the Musqueam fished in the rivers, harvested plants for medicine and food from the land and that there was a temporary village at one time at the point. I explained that the Musqueam community was still a strong, thriving community and that their “village” was now across the river from Richmond. I could tell this was new information for the students and hard for them to understand. At this age, I try and respond to the students in terms of how much information they are able to take in about this and every class is different and has varying levels of background information. We will try and continue building their understanding this spring.

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One of the classes had recently gone out for a shape walk and this is what they focused on in our time outside. They enjoyed identifying shapes but we didn’t get much past that. The teacher and I noted how all the shapes they were noticing were human-created.

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I took the opportunity to have the students look at two trees and see if they could see shapes in them. Although the edges in nature aren’t as clear as in human-made structures/shapes, we did get to students noticing that one tree had a triangle shape while the other was round, or  like a circle. I thought this might inspire some questions. I think next time I will need to structure our time in a way that focuses on wonder – asking the students to wonder aloud, ask questions, make connections and share their thinking.

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One student was able to tell express a big idea that he took away from our time together – that math is more than numbers, it is shapes and patterns too. We made a start! Next time, we will re-visit how we define mathematics, look at the photographs we took and look for some mathematical inspiration in them.


what can your pattern become?

Posted on: April 3rd, 2015 by jnovakowski 1 Comment

Sharon Baatz, a kindergarten&grade 1 teacher at Woodward Elementary has had the Reggio-inspired patterning kit since February. Having already investigated patterns with her students in the fall, the kit gave her and her class a chance to re-visit the big ideas with some fresh materials. Sharon mentioned that her students particularly enjoyed working with the nuts and bolts and that she found the grid and spiral mats really helped her students expand their thinking about patterns.

Sharon sent me some documentation she created about an inquiry that began in her class, emerging from a student’s observation: What can your pattern become?

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I visited Sharon’s class last week and asked them the big question of “what is a pattern?” and again, as for most five year olds it seems, they were able to provide lots of examples of what a pattern was (orange, white, orange, white, orange, white, etc) but had difficulty defining and describing the concept. We struggled through that conversation but  got to some ideas around repeating, alternating and being predictable. I then asked the students to consider the question: What stories live within patterns? I knew this class engaged in the story workshop process so I hoped this question would inspire them. We set out materials on the tables and Sharon followed one of her classroom routines and asked a child to name students to go and choose where to begin their investigations.





Not currently in the kit, but materials I have been using to extend students’ ideas about patterns, is a collection of bare wooden blocks. I find that many students focus on colour or shape when patterning and I wanted to provoke their thinking by having them work with materials that were all the same colour and the same shape.

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We saw some very creative thinking with the materials and the students focused on position and creating height by building with the blocks.

We came together at the end to share and I asked the students if they found any stories. One student commented that she made a zigzag pattern and that could be a path going somewhere. What a great inspiration for a story! This was some new thinking for the students and Sharon explained that she is interested in exploring the idea of math workshop to parallel what she is doing with story workshop.

Our practicum student from Queen’s University interviewed Sharon and captured these reflections:

  • She felt that the openness is good for children because it inspires growth and “encourages different levels of thinking”
  • Sharon liked one of the picture books in the kit and she used it as a “spring board” by showing it to students first to look at patterns
  • Sharon noticed the “social skills that these practices develop” such as working together and sharing
  • Sharon noticed that the “students were very engaged”
  • Sharon liked the “vocabulary the materials and practices encourage” 
  • Sharon likes that the all students can achieve to their own different levels, and that the students often get pushed to further, higher levels of achievement

Sharon’s reflections made me think about our redesigned curriculum and the core competencies. Many teachers have wondered how those will be enacted in the classroom and I think the above examples speak to this. We saw lots of evidence of communication, creative and critical thinking and personal and social development.


thinking about the big ideas of number in grades 2&3

Posted on: April 2nd, 2015 by jnovakowski

On Tuesday, I visited Anna Nachbar’s grades 2&3 class at McNeely with our Reggio-inspired number kit. Anna had attended a session I presented on the project at our Elementary Math Focus Day on December 1 and said she would be interested in investigating the materials with her class. Most of the classes that have used the kits have been Kindergarten or K&1 classes and I really would like us to look at how the materials and Reggio-inspired practices are applicable with older students so this was a great opportunity!

When I looked over the kit list and then the number learning outcomes for grades 2 & 3, I designed some new provocations that would get at some big ideas for those grade levels, using the existing materials with a few little add-ons. Provocations are intended to be inquiry-based and take on different forms in terms of structure. I don’t like to call what we are doing “centres” or “stations” as to me, this brings images of students being grouped or “signing up” to go to certain places to do specified tasks/activities and then rotating through them, often in a timed manner. Instead, I like to invite the students to think about what they are interesting in investigating and begin there (noting if there is a lot of children at one area, asking them to think about how they could handle that problem). Some students may stay with one provocation (that becomes an inquiry for them) for an extended period of time while others may move from area to area or choose different materials to engage with. Aligned with Reggio-inspired philosophy and practices, I believe we have to trust that students are competent and capable of managing themselves and making reasonable choices. When they need support with this, we can coach them through this. One of the things that I have noticed throughout this inquiry project and that many teachers have commented on is that they are astonished by the high levels of engagement and independence that the students demonstrate.

For grades 2&3 the students are learning about 2 and 3-digit numbers with some very big ideas around place value that are foundational at these grade levels. The following provocations were designed to have students play with these ideas and to uncover their conceptual understanding around number.

The following prompt directly guides the students’ inquiry. It asks them to choose three digits from the baskets of different materials (various wood numerals and pebbles) and then to investigate different numbers you can make with the three digits. I also put out mini-clipboards and many of the students enjoyed writing down all the numbers they made. One pair of boys wanted to know the number of  ”combinations” possible and tested their initial theory using various digits. Anna commented that this same problem or investigation was in the math resource they use at McNeely but as teachers, we noticed that there was something about doing this kinaesthetically with the materials that enabled students to come up with most of or all of the combinations (rather than just doing it more abstractedly with just pencil and paper).



Some of the students also began to play with creating very large numbers and were excited to learn how to read them.



Another provocation we set out was a collection of hundred charts, a basket of pebbles and the question: What can you find out about numbers?


At first we noticed some discomfort amongst the students. They weren’t quite sure what to do and looked to us to tell them what to do. With the ball toss back, “what do you think you could do to find out about numbers?” the students began with something familiar – looking for number patterns. This was a start and it will be interesting to see where they take this. Anna has built such a strong sense of community in her class and the students were kind to each other and collaborating so well that even though this type of inquiry-based mathematics was new to them, they were able to adjust and engage.


The following set of materials were presented without any direct prompt. Students used the materials to create games, build and record numbers, etc.


For this provocation, I set out a collection of 1, 2, 3 and 4-digit number cards as well as some pieces of yarn. The invitation was to consider different ways that the students could compare, sort and order the numbers. One student began by making a venn diagram using two pieces of yarn but then wasn’t sure to go with and then said, “Hey, this could be a number line instead!” and lay the yarn out. I watched as two students worked together, taking turns placing the number cards in order along the number line. I watched one student notice a misplaced number and kindly say to her classmate, “do you think this number goes here instead?”



Another provocation asked students “Where do you see numbers in your world? How are they used?” and I had some photographs of addresses out for students to look at with some large wooden numerals. Two boys excitedly explained to me that they were going to make “their” numbers which they explained was the numbers on their homes.


On one table I placed some baskets of glass gems with no direct prompt accompanying them. They were left alone for half of the time (we spent an hour together) and then a group of girls visited the table and began creating patterns. When I noticed this, I asked the students “how do these patterns help you think about number?” This led to a discussion about labelling patterns with number, how many “elements” are in the stem or core of a pattern and then how many terms or groups are within their repeating patterns. This would lead very nicely into an introduction to multiplication, emerging from this experience.


I currently have a Queen’s University student working with me on an alternative practicum and she was able to “interview” Anna a bit and get some of her reflections on the experience.

  • Anna admitted that “I did not know what I was going to see.  I knew the materials were all natural but I did not know how Janice was going to make it grade appropriate.”  Nachbar then added that, “She did make it very grade appropriate,” and was thinking about how she could use similar provocations, materials and practices for other ideas in mathematics, such as multiplication and division.
  • Anna noticed that “All students can be successful and take something from the experience” as the materials and provocations provided multiple entry points, allowing students to enter the activity at their level.  She also believes that these materials and practices would be successful combined with direct teaching.  Nachbar felt that there is “a place for open-endedness” but there must be a balance of this with direct teaching.(Janice’s note: a mini-lesson involving review or teaching of a concept usually proceeds a set of provocations which can be seen as a time to practice, apply and go deeper)
  • Nachbar’s big question was “at what point do you step in a guide the students?” 
    • (Janice’s note: Anna and I had this conversation together and I think like in any situation, as a teacher, you read the situation and respond differently in different contexts and for different students. I think sometimes we feel we need to jump in but often giving the students a little time to struggle lets them figure things out themselves and settle in. When students are off-task and not engaging, we need to think about why and be responsive to that. Sometimes we need to re-direct, pair them up with a classmate, etc).

Anna’s questions about guiding the students also made me reflect on our experiences at the Opal School in Portland and the teachers’ discussion of the ball toss, a metaphor that emerged from Loris Malaguzzi of Reggio Emilia as well as the Harvard Project Zero work around Making Thinking Visible and the use of the “reflective toss” to guide and further inquiry. So I think the timing of providing guidance is important – being responsive, giving time for students to sort things out themselves but also what we say, what language we use is also important. How do we support and extend students’ thinking instead of directing it?

It is so inspiring to think and learn alongside a teacher like Anna who is open to professional inquiry. I am hoping to find some time in my schedule to get back to visit the students at Anna at McNeely this month.