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The Student’s Role in Formative Feedback Cycles Nicole Rigelman, Portland State University Teachers of Teachers of Mathematics Conference September 6, 2013

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Session Overview Engage in a mathematical task and discuss strategies (we’ll return to this later) Examine a research-based vision for instruction and assessment Connect to case of a third grade classroom Classroom Culture Focused on Developing Mathematicians Selecting Tasks to Ensure Quality Data Feedback and Revise Cycles Consider implications for our instruction and assessment practices

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CCSS.Math.Practice.MP3CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others.

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Brownie Sharing Task The sisters decided to each order a different brownie so they could share. Gabrielle ordered a white chocolate brownie and Isabella ordered one with peanut butter chips. Gabrielle said that the piece Isabella gave her was not half of her brownie. Their cuts are shown below. Did they each get a fair share of each other’s brownie? How do you know? Gabrielle’s BrownieIsabella’s Brownie

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Brownie Sharing Task The sisters decided to each order a different brownie so they could share. Gabrielle ordered a white chocolate brownie and Isabella ordered one with peanut butter chips. Gabrielle said that the piece Isabella gave her was not half of her brownie. Their cuts are shown below. Did they each get a fair share of each other’s brownie? How do you know? Gabrielle’s BrownieIsabella’s Brownie

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Common Core Connections 3.G.A.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. 3.G.A.2 3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. 3.NF.A.3

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The Student’s Role in Formative Feedback Cycles

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Solve worthwhile mathematical tasks that are cognitively demanding and connected to appropriate standards. Engage in mathematical practices such as: observe patterns, make use of structure, model mathematics and translate among various representations, engage in a process of conjecturing then justifying and generalizing those conjectures. Students Engage in Mathematical Thinking and Reasoning

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Understand the learning goals /intentions and the criteria for success. Ask genuine questions to deepen their understanding. Critique their thinking and the thinking of others. Engage in self- and peer-assessment. Use feedback as opportunity to grow and improve. Students Own Their Learning

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Classroom Culture Focused on Developing Mathematicians Students made the conjecture “an even plus and odd is always an odd.” The teacher asked the students to decide if they agree with this conjecture.

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Classroom Culture Focused on Developing Mathematicians 8 7

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= 127 (odd) How do you know? Because 27 is odd, and 100 is even How do you know 27 is odd? How do you know 100 is even?

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Classroom Culture Focused on Developing Mathematicians = (odd) How do you know? Because 9 is odd

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Classroom Culture Focused on Developing Mathematicians Because we wanted to move students toward proof that was based on more than just a memorized fact, we decided that the next day’s lesson should focus more on how even and odd numbers look, how the structure can support you in deciding if the sum is even or odd. We wanted students to come to see representations as a potential tool in developing convincing arguments.

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Classroom Culture Focused on Developing Mathematicians

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Developing Classroom Culture through Written Feedback - S’Mores Problem I made s'mores with my daughter and had some leftover chocolate for next time! How much chocolate is leftover? [The teacher showed the students the left over chocolate that was equivalent to 1 ½ or 3/2 chocolate bars.]

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I think being a mathematician is when you are pretty good at math and when you really show work when you are doing something. Like if you are doing fractions or something, you use a model or a number line to show your thinking. It is someone who knows how to learn math well.

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Being picky about what models you are using, how your jumps are on the number line, when you are measuring, starting at zero, and things like that. It is about being confident in yourself and knowing you can do it. It’s not about talent, no matter how smart you are, you can work on it, rely on it more, and do it more. It is not about being the smartest kid in class. It is someone who relies on tools, no uses tools, trying to decide if someone else is right or trying to prove your thinking to them.

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Selecting Tasks to Ensure Quality Data – The Brownie Sharing Task The sisters decided to each order a different brownie so they could share. Gabrielle ordered a white chocolate brownie and Isabella ordered one with peanut butter chips. Gabrielle said that the piece Isabella gave her was not half of her brownie. Their cuts are shown below. Did they each get a fair share of each other’s brownie? How do you know? Gabrielle’s BrownieIsabella’s Brownie

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Feedback & Revise Cycles

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Sometimes I feel kind of sad about that, like when she writes you a question. Sometimes it’s like “Wow, you did it” and sometimes it’s like “You need to rethink this.” It reminds me of being behind in my work because I have to do it the next day. I think it is pretty good. She doesn’t say that we’re wrong, she says rethink. I might need to be more precise or to prove a little bit more. Maybe I need to match my models.

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I like getting feedback because when you get it you get to fix what you did wrong. Well not wrong, you just need to rethink. I like it because you get to know how other people think about your work. It is like having a conversation with the teacher. If I don’t understand something, I can ask a question and she will write me a note or a question that will help me think about it some more. Or we will have a conversation about it.

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I think it helps because when you go to a test she does not give you feedback but by then you kind of know a lot. When you have a chance to fix your stuff you notice what you are missing and I think that helps a lot.

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Implications What are implications for instruction and assessment practices in your courses? as you support teachers? other?

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