One of the many things I love about Twitter is the diversity of the group in which I have the opportunity to interact. Every day, Twitter provides the space for me to move outside of my classroom happenings and connect with others of varying perspectives and insights on teaching and learning. While these perspectives are so interesting to me, if I am being completely honest, they can also be quite intimidating. Not intimidating in the sense that one person’s point of view is “better” than another, but more in the sense that sometimes math conversations go to a place content-wise or philosophically that I cannot even engage. Not because I don’t feel like I don’t belong, but simply because I don’t even know what the heck to say because I don’t understand what they are talking about or it is so far removed from where I am in the classroom, I can’t relate.

The way I feel in those situations feeds my preconceived notions I have about mathematicians. Not the type of mathematicians I would call my students because they are doing great math, but mathematicians as in, that is their job title, you know, those mathematicians. I so admire the way in which they think about math, however given a choice, I would probably shy away from a conversation with them out of shear nervousness of saying something that sounded silly, or even worse, completely wrong mathematically. That was, until I started my work with Illustrative Mathematics.

Throughout my projects with all of the wonderful people at Illustrative, I have truly seen such incredible value for the perspective each and every person, whether a teacher, a math coordinator, a mathematician, or math specialist brings to the work we do in working to improve teaching and learning. From developing tasks, to facilitating professional development, the work is such an amazing collaborative effort in which I learn SO much. During this learning, my confidence in what classroom teachers bring to a math conversation grows, as does my appreciation for our different perspectives.

Most recently, a mathematician at Illustrative, Mike, and I have been working collaboratively on tasks to be reviewed for the IM site. It has been such an amazing learning experience for me. He is wonderfully thoughtful about the math, open to any ideas and/or questions I have and possibly the quickest email responder I have ever encountered:) Throughout our work together, I felt we were on the same page as far as the content of the task as well as in our thoughts about what students would do with the math of the task. I didn’t feel at all like I was “just” speaking from experience and he was talking from this “mathematician world” in which I couldn’t relate, but that we were both thinking deeply about the math and how it looks in a classroom, it was a beautiful thing.

After our first task, I thought to myself how odd it was that we thought so much alike. I was completely anticipating having these eye-opening mathematical revelations after our conversations together. However, during our second task, the revelation(s) came rolling in and the difference in our perspectives was really interesting and valuable.

The task centers around the commutative property of multiplication with fractions in the context of wrapping packages with riboon, 6 x 2/3 and 2/3 x 6. In my classroom, I am so wary of students strictly computing without making sense of problems that I make a conscience effort, probably to almost an extreme, to connect their representations to a context. For example, in the problems above, I really want students to “see” the story for each differently. I want them to see 6 group of 2/3 for 6 x 2/3 and 2/3 x 6 as 2/3 of 6 or an area model with 6 and 2/3 as the dimensions. My biggest concern as a teacher, is the students connecting the problem to the context and then noticing patterns that show commutativity. My questions primarily focus on connecting their representation and notation back to the context. Everything to me is focused on context because of my fear of them number-crunching their way through an algorithm they don’t have a contextual visualization. Did you happen to catch that I care about context in that paragraph:) I even blogged about it here: https://mathmindsblog.wordpress.com/2015/03/29/commutativity-in-fraction-multiplication/

Mike and I both agree all of this contextual work is super necessary and important. This past year, I think my students did a beautiful job seeing the commutative property come out through patterns and repeated reasoning, however, after talking more with Mike about this commutativity, I realized I missed such an important piece. A piece that would have really solidified the commutative property in their work through their representations themselves.

I wanted students to match one of those two equations to a context and develop a more appropriate context for the other, however that just shows they come out to the same answer. In my mind it doesn’t really show how they can be commutative within the same context. I had never thought of that so much until Mike emailed me this statement…

“… if someone arranged the pieces of ribbon appropriately they could argue for either equation. I think that what we are after is to match an expression with some kind of reasoning. In other words, the real question to ask the students is to explain their expression via a picture that accurately models the situation.”

This is the point where I completely wish I could reteach this lesson. I would do everything the same, but add this piece. I would love to see if students could see one representation in another for both 6 groups of 2/3 and 2/3 of 6. Have them defend their reasoning and/or find their reasoning within someone else’s work. That really would have proven to students how the commutative property looks versus just seeing I get the same answer no matter the order of the numbers. Which is kind of how I felt I left it this year.

This has been, and will continue to be, such a wonderful learning experience for me. I SO appreciate the diversity of people I have worked with at Illustrative as much as I appreciate the wonderful mix of people I get to learn from on Twitter. It is enlightening to me that as open and addicted as I am to learning, there are still so many things that I have a classroom perspective on that can be improved and extended through conversations with people who I may typically have shied away from in person. Knowing they appreciate my perspective is such a wonderfully empowering thing for me as a learner. Thank you to all involved in my journey!

howardat58Fascinating, especially after reading the previous post. One of the problems with the teaching of fractions is the “rush to symbols”. Another is introducing fractions as “parts of a whole”. Really what one is doing is using a fraction as a description of a “part”. There is a BIG difference here. It’s the same with proportions. Proportions can be described by fractions. I am sending you my thoughts on fractions (actually developed as fractions for adults). The first two pages (of 9 !) are crucial, no fraction symbols at all.

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mathmindsblogPost authorThank you so much Howard and I couldn’t agree more on all accounts. I love your description of fractions as a description of the part. It is such a great way to explain it because so many teachers explain fractions only in terms of parts of a whole. I look forward to reading your email!

Thanks!

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howardat58Another thought:

Try six rectangles end on, horizontally, representing the six 2/3 length pieces. Draw a line across the page, 2/3 of the way up, cutting each rectangle.

Well, that’s 2/3 of six as well, won’t it do?

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mathmindsblogPost authorYes! This is exactly the piece that Mike was talking to me about, connecting the two representations. Such a concrete way of seeing the commutative property in action!

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