Back in January, Michelle Hikida and I introduced the Reggio-inspired patterning kit to her grades 2 & 3 class at Diefenbaker and we considered the affordances of different materials to support mathematical thinking and inspire inquiry. A blog post about this experience can be found HERE.
Later in the term, Michelle approached the concept of fractions in the same way, laying out a variety of materials and asking students to show what they knew about fractions. What happened surprised her and caused some reflection. Instead of representing their understanding of fractions with the loose parts and math materials, they represented the symbolic notation of fractions. With discussion, Michelle realized this is what they knew about fractions, that they didn’t understand the concept but were familiar with the symbolic notation.
For example, students initially represented fractions this way:
This made Michelle think back to the experience when she introduced patterning. Students in grade 2 and 3 have previous school experiences with patterning and have a place to start when demonstrating their understanding. For fractions, although students may have had informal experiences at home, the concept of fractions is not formally introduced until grade 3 in our curriculum. Michelle spent some time working with loose parts and math materials to use an inquiry approach to develop understanding of fractions. By asking questions like “What is a half?” and “How could you show what 3/4 means?” the students were able to develop and represent a conceptual understanding of fractions using loose parts and math materials.
I visited the class a few weeks later and students had already made big jumps in their conceptual understanding and were able to represent fractions both concretely and pictorially, connecting to the symbolic notation.
Another example of representing mathematical thinking with loose parts is from the grade 3 class at Quilchena. Although we had also looked at creating representations of what multiplication and division meant, for this class students were given loose parts to represent specific multiplication equations. The following example shows that the student understands that 5×2=10 by showing five groups of 2. If the student had used the loose parts to represent the equation by making a 5 and then a 2 and adding “symbols” made of other materials, it would not show evidence of conceptual understanding, just a representation of the equation.
I think this is where as educators, we need to be keen “noticers” when students are using materials and consider the following questions:
How are student using the materials?
What are the materials offering the students (or not)?
Do some materials have more affordances than others for specific concepts?
Are the materials supporting students’ thinking and understanding?
Are our questions or provocations supporting thinking and understanding?
What do students need in order to use loose parts successfully? What do we need to do as educators?
For me, this is a matter of responsiveness and awareness. To be responsive to what we notice in our students, we need to take time to observe, notice, and be curious about their learning but we also need to be aware and knowledgable about the mathematics that the students are investigating so that we can respond and provoke their thinking.