As a teacher, curiosity around students’ mathematical thinking was the driving force behind the teaching and learning in my classroom. To better understand what they were thinking, I needed to not only have great, accessible problems but also create opportunities for students to openly share their ideas with others. It only makes sense that when I learned about routines that encouraged students to share the many ways they were thinking about math such as Number Talks, Notice and Wonder, and Which One Doesn’t Belong?, I was quick to go back to the classroom and try them with my students. It didn’t matter which unit we were in or lesson I had planned for that day, I plopped them in whenever and wherever I could because I was so curious to hear what students would say. Continue reading

# Category Archives: Illustrative Mathematics

# Explicit Planning vs Explicit Teaching

*Planning is like…..*

How would you finish that sentence?

As a facilitator, I use this sentence starter to open Illustrative Mathematics’ 5 Practices Professional Learning. To be completely honest, when I designed the PD I was a little hesitant of using it because I was nervous it was opening a can of worms within the first 5 minutes of the day. I am, however, always surprised with all of the beautiful analogies participants share and feel challenged each time to come up with something new and better than the one I used in previous sessions. When I first delivered this PD, I started with analogies like a marathon or really hard workout – something that is exhausting, a lot of work, but ends with something I take pride in. While these analogies were accurate representations of how hard I think lesson planning truly is, I was continually unhappy with where students’ ideas fit into my analogy.

My most recent sentence was this…

*Planning is like putting together a puzzle. *

When sharing my reasoning with participants for the first time, I included a lot of beautiful words around mathematical connections but in the middle somewhere I used the phrase *“making connections explicit”* in relation to the puzzle pieces and saw an immediate reaction from a few people in the room. Of course, I had to pause and ask, *“Was it the word explicit?” – *answered by many nods in the room.

For a long time, the word explicit in relation to teaching held a negative, cringe-worthy connotation for me as well. If ever asked to paint a picture of what *explicit* teaching looks like in the math classroom, I would describe scenarios in which a teacher is either at the board telling students how to solve a problem or showing a struggling student how to solve a problem because they are stuck or “taking the long way there.” To me, being explicit meant telling students a way to do something in math class – typically in the form of a procedure.

Through teaching a problem-based curriculum [Investigations], designing and implementing math routines such as number talks, and reading Principles to Actions, 5 Practices and Intentional Talk , I realized that I was guilty of making mathematical ideas explicit every day in my classroom, but not in the way that made me cringe.

I was explicitly planning, not explicitly teaching.

To me, those two phrases indicate a big difference in how I think about structuring a lesson. I have found when teaching a problem-based curriculum, it is easy for ideas to be left hanging and important connections missed, forcing me to explicitly teach an idea to ensure students “get it” before they leave the class period without any understanding of the mathematical goal for the day. Many days, I would find myself frustrated because students would completely miss the point of the lesson, however now I realize this was because I was expecting them to read my mind of what I wanted them to take away from the problem. On the flip side of that coin, however, not teaching a problem-based curriculum and explicitly teaching students how to do the math in each lesson is not an option (and is a topic that could be its own blogpost).

This is exactly why I find the 5 Practices framework invaluable in planning. The framework forces me to continuously think about the mathematical goal, choose an activity that supports that goal, plan questions for students toward the goal, and sequence student work in a way that creates a productive, purposeful discussion toward an explicit mathematical idea. I have learned so much using this framework over and over again in planning for my 5th grade class, collaborating with other teachers and coaching teachers across different grade levels.

Explicit planning is how I would describe the new, open education resource (OER) by Illustrative Mathematics. As a part of the writing team, I explicitly planned warm-ups such as number talks and notice and wonder activities to elicit specific mathematical ideas that play a purposeful role in the coherent plan of the lesson and unit. But not only are the warm-ups explicitly planned, but each lesson and unit tells a mathematical story in which students arrive at a specific mathematical landing point. While they may not all arrive at that landing in the same way, the problems and discussions are structured to ensure students do not leave the work of the day without any idea of what they were working toward.

While I would love to think my blog posts paint a clear picture of explicit planning, I am not that naive. So, what does explicit planning look like in a 5 Practices Framework?

This lesson from Grade 7, Unit 2, Lesson 2 from Illustrative Mathematics’ Middle School Curriculum is one of many in the curriculum. (All images are screenshots from the online curriculum that is linked at the bottom of the post)

*Practice 0: Choosing a Mathematical Goal and Appropriate Task*

With the goals in mind, the lesson begins with a notice and wonder warm-up that engages students in thinking about tables, followed by two activities that build on those ideas and support the mathematical goals. While both activities demonstrate explicit planning, I am focusing on one for the sake of space.

**Practice 1 and 2: Anticipating & Monitoring**

**Practice 3 & 4: Selecting & Sequencing**

*Practice 5: Connecting*

*So..Planning is like putting together a puzzle. It is hard, takes time, and is sometimes difficult to figure out where to start. We know all of the pieces connect in the end, but making a plan for all of those pieces to connect takes an understanding of the final picture – the goal. There will be missteps along the way and some parts will take longer than others, but we know it is important to carefully connect each piece to another as one missing piece will leave unconnected ideas and the final picture unfinished. As you work alone, the way the pieces connect to form the final picture may not always be obvious, but as others help us see the pieces in different ways during the process, connections become explicitly clear and the final picture is something in which you can take a lot of pride.*

The ‘others’ in my teaching journey have helped me see a difference between explicit teaching and explicit planning. Through explicit planning I have seen the importance in understanding the mathematical goal in a way that enables me to structure activities and lessons that enable students to make important mathematical connections through their own work and discussions. It is so exciting to see IM’s curriculum be a model for how I think about explicit planning in such a coherent, purposeful progression.

Link to Illustrative Mathematics 6-8 Math Curriculum.

Link to the 7th Grade lesson featured in this post.

# Extending an Area Task

I just love when students are so excited to extend an activity! During the Notice and Wonder portion of this lesson, a lot of students wondered why those four letters were the ones given. Was it because they are at the beginning of the alphabet? Is it because they have the same area? What would happen with other letters?

Today, Mrs. Sharp gave them the chance to play around with others letters. She asked them to design their own letter and find its area. It is so interesting to see their choice of letters, the way they chose to decompose the shape and their math work all around it. Many of them made multiple shapes because they just wanted to keep going…that is always so AMAZING to hear!

I also see all of this being SO helpful when they find volume of figures composed of two non-overlapping rectangular prisms in 5th grade!

Here are just a few of the creations they came up with in class today:

# A Teacher & Mathematician Mash Up

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!