I do think that justification of their solutions gets at some important points regarding mathematical reasoning, but would like to move the discussion to center around that exact feature--the

*reasoning*.

Rather than piecing together the pieces of isolated reasoning (which I still think has value), I want to see a collective reasoning emerge through the discussion. Students don't "think-pair-share" their solutions; students bring their conceptualizations and reason to a collective understanding. There is something about the power of this collectivity that makes the learning unique. (And I am insanely interested in how it differs, why it differs, and if it is viable in a classroom).

To try and harness this type of reasoning, I created (with my students) a series of "Fraction Squares" inspired by a keynote given by Ilana Horn and SUM2015. They all involve a dissection of a square using straight lines.

I project a Fraction Square, shade a section, and ask, "

*What fraction of the square is shaded?*"

I moderate the discussion and entertain all solutions. It isn't long until the discussion no longer runs through me. The students turn, pull out paper, run to the board, and justify their thinking.

As they talk about slicing the sections into pieces, I do my best in interject precise vocabulary. For instance, a student might say, "A half of a half is a quarter", and I may probe this to elicit which operation they are really talking about. I don't push the standard algorithm of multiplying fractions, but write number sentences beside their quotations.

This way, the vocabulary and notation is slowly imbibed into their collective action. Students start to talk about the four operations (instances of division by fractions are rare), and the number sentences act as a bridge to conceptualize the standard fraction algorithms. It provides an active visualization of processes like creating a common denominator. The picture is not presented and explained; the picture is presented and then

*acted out*through collective discussion. In a sense, it becomes personal.

Some groups may focus on certain lines of argument, and others find them completely disinteresting. It is the job of the teacher to remain fluid, keep end goals in mind, and network the collective.

Some other prompts to use:

*Which section is the easiest to find?*

*How many different sections exist and what are their values?*

*If the total area was "x", what is the area of the piece?*

*Can you find all pieces that are exactly "1/x" of the total?*

*Can you find a piece that is double / half of the section?*

NatBanting

Davis, B. (1996)

*Teaching mathematics: Toward a sound alternative*. New York, NY: Garland.

Great ideas. I use to use an overhead with coloured opaque tiles that were arranged in a square. Not as easy to make different forms. This offers so many different options. The great thing is the ability to share with other students and share their thinking. Collaborative groups for sharing their ideas and thoughts is a great way to help students work through their thinking - having them share helps all students especially with fractions. Thanks for sharing your work and what you are doing!

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