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).

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

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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?

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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.

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

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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?”

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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.

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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.

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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.

~Janice

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