Category Archives: Professional Development

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 Actions5 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

Screen Shot 2017-08-25 at 10.13.27 AM.png

Lesson Learning Goals

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.

Screen Shot 2017-08-24 at 10.29.24 PM.png

Task Statement

 

Practice 1 and 2: Anticipating & Monitoring

Screen Shot 2017-08-24 at 10.42.08 PM

Activity Narrative

 

Practice 3 & 4: Selecting & Sequencing

Screen Shot 2017-08-24 at 10.43.59 PM

Activity Synthesis

 

Practice 5: Connecting

Screen Shot 2017-08-24 at 10.52.01 PM

Activity Synthesis

 

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.

Asking Better Questions

I am sure we have all seen it at one time or another – those math questions that make us cringe, furrow our brow, or just plain confuse us because we can’t figure out what is even being asked. Sadly, these questions are in math programs more often than they should be and even though they may completely suck, they do give us, as educators, the opportunity to have conversations about ways we could adapt them to better learn what students truly know. These conversations happen all of the time on Twitter and I really appreciate talking through why the questions are so bad because it pushes me to have a more critical lens of the questions I ask students. Through all of these conversations, I try to lead my thinking with three questions:

  • What is the purpose of the question?
  • What does the question tell students about the math?
  • What would I learn about student thinking if they answered correctly? Incorrectly?

Andrew posted this question from a math program the other day on Twitter….

Screen Shot 2017-02-09 at 7.13.35 PM.png

I tried to answer my three questions…

  • What is the purpose of the question? I am not sure. Are they defining “name” as an expression? Are they defining “name” as the word? What is considered a correct answer here?
  • What does the question tell students about the math? Math is about trying to interpret what a question is asking and/or trick me because “name” could mean many things and depending on what it means, some of these answers look right. 
  • What would I learn about student thinking if they answered correctly? Incorrectly? Correctly? I am not sure I even know what that is because I don’t know what “name” means in this case. Is it a particular way the program has defined it?

On Twitter, this is the conversation that ensued, including this picture from, what I assume to be, the same math program:
Screen Shot 2017-02-09 at 6.56.47 PM.png

When a program gives problems like this, we not only miss out on learning what students know because they get lost trying to navigate the wording, but we also miss out on all of the great things we may not learn about their thinking. For example, even if they got the problem correct, what else might they know that we never heard?

The great thing is, when problems like this are in our math program, we don’t have to give them to students as is. We have control of the problems we put in front of students and can adapt them in ways that can be SO much better. These adaptations can open up what we learn about student thinking and change the way students view mathematics.

For example, if I want to know what students know about 12, I would just ask them. I would have them write in their journal for a few minutes individually so I had a picture of what each student knew and then would share as a class to give them the opportunity to ask one another questions.

After I saw those the problem posted on Twitter, I emailed the 2nd grade teachers in my building and asked them to give their students the following prompt:

Tell me everything you know about 12. 

20170208_092159 (1).jpg

Ms. Thompson’s Class

IMG_5487 (1).jpg

Mrs. Leach’s Class

IMG_5381.JPG

Mrs. Levin’s Class

Look at all of the things we miss out on when we give worksheets from math programs like the one Andrew posted. I do believe having a program helps with coherence, but also believe it is up to us to use good professional judgement when we give worksheets like that to students. While it doesn’t help us learn much about their thinking it also sends a sad message of what learning mathematics is.

I encourage and appreciate conversations around problems like the one Andrew posted. I think, wonder, and reflect a lot about these problems. To me, adapting them is fun…I mean who doesn’t want to make learning experiences better for students?

Looking for more like this? I did this similar lesson with a Kindergarten teacher a few years ago. Every time I learn so much and they are so excited to share what they know!

Number Talk Karaoke

It is always so fun when I have the chance to hang out with my #MTBoS friends in person! This summer Max was in town, so I not only got to have lunch with him but also meet his amazing wife and puppy!  Of course, during lunch, we chatted a lot about the math work we are doing with teachers and some of the routines we are finding really valuable in their classrooms. From these two topics of conversation, Number Talk Karaoke emerged.

We both agreed that while Number Talks are invaluable in a classroom, it can be challenging to teach teachers how to use them in the classrooms. As much as we could model Number Talks during PD and show videos of them in action, it is still not the same as a teacher experiencing it for themselves in their classroom with their students. There is so much to be said for practicing all of the components that are so important during the facilitation with your own students.

That conversation then turned into two questions:  What are these important components? and How do we support teachers in these areas?  We discussed the fact that there are many books on mathematical talk in the classroom to support the work of Number Talk implementation, however the recording of student explanations during a Number Talk is often left to chance. What an important thing to leave to chance when students often write mathematics based on what they see modeled. We brainstormed ways teachers could practice this recording piece together, in a professional development setting, where students were not available.

Enter Number Talk Karaoke.

During Number Talk Karaoke, the facilitator:

  • Plays an audio recording of students during a Number Talk.
  • Asks teachers to record students’ reasoning based solely on what they hear students saying.
  • Pair up teachers to compare their recordings.
  • Ask teacher to discuss important choices they made in their recording during the Number Talk.

Max and I decided to get a recording and try it out for ourselves. So, the next week, I found two of the 3rd grade teachers in my building who were willing to give it a go!

They wanted to try out the recording piece themselves, so they asked me to facilitate the Number Talk. They sat in the back of the room, with their backs to the students and SMARTBoard so they could not see what was happening. All they had in front of them was a paper with the string of problems on it.

Before seeing our recording sheets below, try it out for yourself. In this audio clip of the Number Talk, you will hear two students explain how they solved the first problem, 35+35. The first student explains how he got 70 and the second student explains how he got 80.

Think about:

  • What do you think was really important in your recording?
  • What choices did you have to make?
  • What question(s) would you ask the second student based on what you heard?

The talk went on with three more problems that led to many more recording decisions than the ones made in just those two students, but I imagine you get the point. I have to say, when I was facilitating, I tried to be really clear in my questioning knowing that two others were trying to capture what was being said. That makes me wonder how this activity could be branched out into questioning as well!

Here was my recording on the SMARTBoard with the students:

Screen Shot 2016-11-13 at 9.12.41 AM.png

Here are the recordings from the two teachers in the back of the room:

screen-shot-2016-11-13-at-9-13-22-am

screen-shot-2016-11-13-at-9-12-58-am

We sat and chatted about the choices we made, what to record and how to record certain things. We also began to wonder how much our school/district-based Number Talk PD impacted the way we record in similar ways.

Doesn’t this seem like a lot of fun?!? It can be done in person like mine was, or take the audio and try it with a room of teachers, like Max did! <– I am waiting on his blog for this:) Keep us posted, we would love to hear what people do with this!

 

Establishing a Culture of Learning …The First Hour

Every year, we as teachers work so hard to establish a safe, open place for our students to learn. My goal in moving out of the classroom year and into a math specialist role is to also establish this same culture among our staff. A culture where teachers talk about instruction, math problems, and student ideas, feel ownership in their lessons and the lessons of others, and can comfortably visit one another’s classrooms. It becomes a norm. It is not easy and definitely cannot be done alone. I am SO incredibly fortunate to have a wonderful principal, Jenny (@PrincipalNauman) and district supervisor (@EducatorKola) who support the vision and are always open to new ideas, a great ELA counterpart Erin (@EGannon5) who helps me focus and thinks about the important details I miss in my excitement about things, incredibly caring, motivated colleagues who always want to grow and learn, and all of the amazing educators in my face-to-face and online (#MTBoS) networks who I mention throughout this post.

Yesterday was Erin and my first opportunity to talk with teachers. We only had one hour to work with the full staff, so we had to truly prioritize and make the most of every minute! We decided it was most important to set the tone for the year and our work together with the teachers. We wanted to begin establishing a culture of learning. The best part was, we were not starting from scratch! Our staff is so wonderfully open to new ideas and really took Number Talks and ran with them over the past few years, however there is always room to grow and improve upon what we were already doing. PLCs are part of that room to grow. While they participated and did everything asked of them, teachers were not feeling that time was based on their individual needs as much as it should be. Being one of those teachers last year, I put myself in that group.

Establishing a Culture of Learning

Establishing a Culture of Learning (2)

Instead of telling them what a learning culture could/should look and feel like, we wanted them to experience and reflect on it. What better way to do that than Talking Points? (shout out to Elizabeth @cheesemonkeysf) We designed the Talking Points to give teachers a range of ideas of how they could be used, whether around content specific statement or ones around mindset.

Establishing a Culture of Learning (3)

I have never been in a PD where Talking Points are not a hit during the activity itself, but the reflection afterward is twice as valuable! We asked them how this activity would promote a culture of learning in a classroom. We tried to quickly list ideas as they responded so the list doesn’t truly capture the appreciation teachers had for students talking and listening to one another!

IMG_0551

When talking to my colleague Faith (@Foizym) about our plan for the hour, I really stressed how I wanted to make my work with teachers valuable this year. I wanted them to want to talk math with me and want me in the classroom and not see me in any type of evaluative role, I wanted our work together to be about their needs in order to best meet the needs of their students. She suggested having them write goals for themselves and their students. So, we asked them to complete these questions to know what each of their goals were…

Establishing a Culture of Learning (5)

We got amazing answers that really spoke to the thoughtfulness of our staff. I would love to post a few but I have to ask for some permissions first:)

Now we moved into how we were visualizing this culture permeating through our work together. Knowing we were introducing Learning Labs and Teacher Time Outs to them soon, we wanted to have them brainstorm words they associated with the word “Lab” and “Time Out” to set the stage. These slides did not have the words/ideas around it when they saw it, we put those up after they brainstormed.

Establishing a Culture of Learning (6)IMG_0549

Establishing a Culture of Learning (7)IMG_0550

Now we described our shift from PLCs to “Learning Labs” and the use of Teacher Time Outs. If you have not heard of Math Lab or Teacher Time Outs, I will point you to Elham Kazemi (@ekazemi) and her University of Washington peeps who are doing AMAZING work with this. Here is her ShadowCon speech that gives a wonderful description. Elham has been so generous in thinking this through with me and has given me wonderful advice, much of which I will continue to need I am sure!

Establishing a Culture of Learning (8)

Establishing a Culture of Learning (10)

Establishing a Culture of Learning (9)

I hope we captured it as she intended, but sadly at this point we were running out of time. There were many questions about the timely structure (that we honestly are still trying to hammer out) but overall everyone was really excited about this work! We received so many positive comments and offers to be the first to try out whatever we wanted to do!

I left completely excited about this work…even more excited than I was to start, if that is even possible! Once Erin and I work through the time constraints and the crazy schedules we know everyone keeps as teachers, I cannot wait to see the work that awaits all of us!

-Kristin

My Questions Around Professional Development w/Video

When I read this blog post by Grace, and the comments that followed, I noticed some things…

1 -The wonderfully open, honest way in which Grace put herself out there and responded to each of the comments.

2 – The amount of incredibly thoughtful and thought-provoking comments.

3 – The community desire to have more of these videos for us to have discussions around.

4 – I reflect and learn so much through these interactions.

*At this point of reading, If you do not already follow Grace’s blog, you must do that ASAP*

My noticings led me to these questions around the types of video we, as a math community, would like to have available for either individual or group professional learning experience:

1 – What time range do you prefer when watching an classroom video clip? Is it different in a professional developmet setting vs at home?

2 – Do you like an open Notice/Wonder format when watching/discussing a video or prefer having a “focus question” when watching/discussing?

3 – What focus questions would be most helpful for you to either think about or discuss after watching a video?

4 – What makes you want to comment on a video or blog after watching/reading?

5 – What makes you NOT comment on a video or blog after watching/reading?

If you have any thoughts, answers, or suggestions to any of these questions, I would love your thoughts here: Google Form

Thanks so much!

Kristin

Number Routine PD: What Do I Know About…

My colleague Nancy and I facilitated a K-2 afternoon professional development session yesterday afternoon. The 2.5 hour session was with a wonderful group of teachers from across our state who we are fortunate to work with several times over the course of the school year. Our major focus over the course of this school year centers around connecting arithmetic to algebra based on a book by Virginia Bastable, et al, that I blogged about here: https://mathmindsblog.wordpress.com/2014/11/20/articulating-claims-in-math/ I thought blogging about this experience would be helpful for any of our teachers who could not attend and for any others who facilitate PD.

I find planning for professional development is much like planning for the classroom. Many of the same questions arise:

What content will be engaging and relevant? (especially being an afternoon session when everyone is winding down on a Friday)

What is the trajectory of the content?

Where are they? Where are they heading?

What questions or prompts will encourage conversation?

When are points for table conversation? Whole Group conversation?

How will be know where they are in terms of the content when they leave us?

How will we follow up?

After much planning, videoing, and organizing this was the flow of the afternoon:

We opened by getting into grade level groups to discuss the homework from last month, doing a group planned Number Talk with their students. They used this form to plan together and brought back recording sheets of their work to discuss these two questions:

ntp nt1With the number talk being planned by the group, I felt a sense of ownership over the results in the classroom and, really, who doesn’t like talking about all of the wonderful things our students say during a number talk?

We continued with a quick recap of last month’s session on the book, “Connecting Arithmetic to Algebra” to plant the seed for our routine of focus that day, What Do You Know About….?

 21 3Now into the really fun stuff! Working with a Kindergarten teacher in my school (@jennleachteach) who is also a part elementary pd group, we planned and videoed a math routine called “What Do You Know About 15?” in Jenn’s class.

We mixed the grade level PD groups up at this point so there was a range of K-2 teachers (and a few math coaches) in each group.  They got a blank planning sheet to brainstorm what they think the planning would look like for this routine in a Kindergarten classroom in January. It was great conversation, with the Kindergarten teachers being the experts at each table. I thought this was such an interesting dynamic since we often tend to pose a mathematical idea and ask what previous understandings K-2 need to build to get there, however, with this opportunity, it was starting in the opposite direction and really focusing on what Kindergartners know at this point of the school year.

4After they predicted what our planning sheet would look like, Nancy brought 6 teachers up to act as students in a fishbowl enactment of the Number Routine. The other teachers in the room were observers focusing on two particular aspects of the talk, what you notice about the teacher recording and what you notice the “students” noticing. Importance of recording was a previous topic in an earlier pd, so we wanted to be sure that resurfaced. Nancy did the routine with the teachers and  we came back as a group to discuss the observations of our focus questions. Our discussion also touched on the use of the talk moves she used to clarify and illustrate student thinking.

We then watched Jenn’s Kindergarten class do the same exact Number Routine, focusing now on the follow up piece of the planning sheet. What did they notice the students noticing? I wish I had permissions from everyone because Jen did a beautiful job in facilitating the talk and her students said some amazing things. We also took a look at the planning sheet that Jenn, Nancy and I had done for this routine. Here is the planning sheet and anchor chart that arose from the talk:

IMG_8758

5

As a group we discussed what they noticed the students noticing that could lead to future “claims” in their classroom. Teachers noticed such things as, “We can count by fives to get to 15” “It is three fives” (of course I am thinking about groups of and multiplication right there!) “A teen number is a group of ten and some more” “Looking at equality with related equations” and “The 1 means one ten”

Jenn then gave students “random” journal entries to see how students were thinking about the numbers after the talk. To differentiate, we decided to give students 12, 19, or 21 depending on where we thought their entry level was into this thinking. After students completed the journals they chatted with someone who had a different number, to talk about their ideas.” Here are the student samples our PD group looked at and discussed:

Photo Jan 08, 5 35 21 PM Photo Jan 08, 5 35 18 PM Photo Jan 08, 5 35 15 PM Photo Jan 08, 5 35 12 PM Photo Jan 08, 5 35 08 PM Photo Jan 08, 5 35 05 PM Photo Jan 08, 5 35 03 PMWe ended with Virginia’s conclusion slide about Connecting Arithmetic to Algebra and our homework for the group:

6 7We also gave an Exit sheet to help us in future planning. We got some very useful information as to where the teachers feel they are. I am very excited to hear about everyone’s journey back in their classrooms next month!

Photo Jan 10, 8 53 02 AM Photo Jan 10, 8 52 41 AM Photo Jan 10, 8 52 29 AM-Kristin

Lesson Study: Teaching “Take 2”

Every week, I read so many wonderfully open and honest math blogs from my tweeps, the majority of them found on this list. The blogs span across grade levels, mathematical content, teacher experience, and more impressively, the world. Whether it is a good, bad or ugly lesson, after reading the blogger’s reflections and colleagues’ comments, I am always left with the feeling that if, given the chance to do that same lesson just one more time, there would be significant improvement. Whether it be the organization of the lesson, management of the materials, questioning of the students, sharing out of responses, the task itself, or any of the other countless components of just one math class period, sometimes we all just need someone to say “Cut” on Take 1 and allow us the opportunity for a Take 2. With multiple math classes a day, we often get the chance to adjust a lesson between class periods, however there is not a second chance on that same exact lesson until the following year, nor a significant amount of time to make dramatic changes. I am not saying that we don’t revisit, learn and improve from that point on, however, how amazing would it be to erase a lesson that didn’t go “quite as planned” from a student’s memory and make it even more meaningful for them on our second take? Wouldn’t it be great to answer all of our “What ifs”?

When selected to participate in the project I blogged about here, I had the idea that it could work as a type of lesson study. Since Alicia and I were both 5th grade teachers working on fractions at the beginning of the year and planning a common task, I thought it would be interesting to see how our work together could go above just collaborating around the lesson development to actually creating a “perfect lesson”…or as close as we could possibly get to it. This lesson study would be unique in the sense that all of our work and observations would have to be virtual due to the distance between us. After all of our planning around the 5 practices, our team of Jody, Chepina, Alicia and her math coach, Jennie, decided that I would teach the lesson first, they would all observe the video (through Teaching Channel Teams), we would look at my student work and from there make adjustments for Alicia’s lesson. Being super critical of my own practice in general, it was fairly simple for me to make suggestions for improvement.:)

THE TASK: Part 1: The 5th graders want to raise money for their overnight camping trip by selling cornbread during the school district Chili Cook-Off contest. All of the cornbread pans are square. The first customer, Mrs. Farmer, wants to buy 1/4 of a pan of cornbread that is 1/3 full. What fraction of the whole pan does she want to buy? Part 2: The next customer is the elementary school principal.  He wants to buy 5/6 of a pan of cornbread that is 1/2 full. What fraction of the whole pan does he want to buy? Each part also had a part b asking if the pan costs $12, how much would they pay for their piece.

MATHEMATICAL GOAL: Students will develop mathematical generalizations connecting previous understandings of whole number multiplication to multiplication with fractions. (Relational understandings)

Just some of my 5 Practices planning for the task:

photo 1 photo 2 photo 3 photo 4 photo 5

With that little bit of background, here is a look into the lesson and the adjustments we made to improve…

My Class: I opened the lesson with a multiplication Number Talk. It was our hope that through the progression of problems, the area model would arise to allow for connections to our task,  however in a classic case of anticipating gone bad, there was no area model to be found that day. Great thinking around the multiplication, so I wasn’t disappointed in that, but no array.

Alicia’s Class: After collaborating around the video and my reflection, we decided to start with a number talk on fraction of a whole number. Both of our classes have been working on that and thought it may put students in more of a “fraction mindset” of taking a part of something before beginning the task. We designed the progression: 1/2 of 16, 1/4 of 16, 1/8 of 16, 3/8 of 16 in hopes of pulling out thoughts such as: dividing by 2 is the same as taking half (and 4 for the fourth), a half of a half is a fourth (and the same reasoning for the 1/4 and 1/8), as well as big ideas around equivalence and decomposition. Perfect change.The students shared all of the ideas we wanted to bring out, even as far as pushing the decomposition of 3/8 from 1/8+1/8+1/8 to 3 x 1/8. Not in that exact notation but as 1/8 of 16 = 2, so 3 x 2 = 6.The only thing I wish I had seen (simply because I love when students make connections to previous problems in a NT progression) is 1/4+1/8= 3/8, so 4+2= 6.

Timing was great and the lesson was improving already…..

My Class: As I read the problem aloud and used some listening techniques I learned from @maxmathforum, students had individual think time and moved into working in a group. The first thing I noticed in my students who struggled with entry into the problem was the wording of the problem itself, “1/4 of a pan 1/3 full” muddled the whole and was confusing for them. Also, and I have mixed feelings about this, but the first part allowed for students to get the correct answer by subtracting 1/3 – 1/4 to arrive at 1/12. This is not something we had even thought about. I love the conversation that arises from this, especially because it will not work in the second part, however I thought maybe it did lead us astray from our goal for the lesson.

Alicia’s Class: Alicia launched the lesson with an actual pan of cornbread to show the class, most impressively homemade:) It gave the students a nice visual for their models/representations and I think offered access to those without entry. We changed the wording of the task so her task read, “Mrs. Farmer, walks up to buy cornbread and the pan is 1/4 full. She wants to buy 1/3 of the remaining cornbread. What fraction of the whole pan does she want to buy?” and the same type of wording changes for the second part. We also changed the order of the fractions in the first part so subtraction would not happen them upon the correct answer and in this case be an unreasonable answer of – 1/12. Having not anticipated subtraction, which seems SO obvious now…duh, it was nice to be aware that it may show up in her class and adjust accordingly. Again, I love that conversation of why it is not subtraction and will definitely revisit, but for the sake of our goal, this was much improved.

After part 1, we planned to do a quick share of strategies.

My Class: During the share, I selected and sequenced three papers. My initial thought was to share two correct representations and one with a sticking point I noticed showing up on numerous papers. I chose two students to share who had the array correct, but cut a bit differently and then a subtraction student with two separate pans drawn (one with 1/3 and one with 1/4). My hope was to draw out, from the class, that we were just dealing with one pan and get them thinking about if that made sense. However, the share went longer than anticipated so I decided to leave that out there for them to think about and move around to my few who were sticking with subtraction individually with some strategic questions about the whole.

Alicia’s Class: After seeing the lengthy share in my video, we decided to have Alicia choose three students,with correct representations and different labeling/cutting, with the third being a student who is having some numeric notation around where he/she is seeing multiplication. Alicia’s final sharer ended with his representation and numeric notation showing that he multiplied the numerators and denominators to get the answer.  He ended his share wonderfully by saying you not taking 1/3 of the whole you are only taking 1/3 of the 1/4. After talking to Alicia and Jennie afterwards, this student has had some outside experience with multiplying fractions in terms of the algorithm, however struggled a bit to connect it to the why. It was nice to have him share and explain his representation.

Now both classes ventured into part 2 of the task.

My Class: I actually enjoyed this part of my lesson, and was excited to see many of my students up and using the fraction bars to create proofs for their tablemates. The biggest obstacle in my class seemed to be naming the piece. I found that after working through the first part and seeing the share, students were more comfortable with what was happening in the problem and could shade the piece the principal was getting. Many were saying the answer was 5/6, 10/12 or 5 pieces. All correct answers, but that fraction of what whole or 5 pieces the size of what? They had great discussion (even a bit of a heated argument between friends) about whether the answer was 5 or 5/6. I love that they left for lunch still arguing about it…nerdy fabulousness. I did my share with students whose representations were a progression of a student who cut the whole pan into twelfths, another who was just half cut but visualized the other twelve and one who solved it backwards at first, 1/2 of 5/6 and their partner explained it to them and redid it. I thought it was a beautiful picture of the commutative property, that even though the order doesn’t matter in the solution, the picture changes.

BzJts81CcAIpauX8

Alicia’s Class: I felt this part was very similar between the two classes. Her share was nice (and I wish I had done it) because she brought up one of her students who had it wrong to start and was brave enough to get up and say what she tried that didn’t work and then what worked for her the second time after talking with her group. It was beautiful and connected a lot of learners who tried it the same way the first time. She opened it up to the class for a few comments/questions after each sharer. The students were very nice in giving positive feedback as to how easy to understand the representation was and stressed that the labeling made a big difference.

I am thinking it would help in share outs to focus the students more on what we want them to be noticing, as my buddy @maxmathforum says often. For example, since our goal was to bring out the connection to multiplication, tell students that as others are sharing, be looking at their representation and for the operations they used and why they are using them to get their answer. I would love for it to explicitly come out WHY the numerators and denominators are multiplying and the connection between dividing into 6 parts is the same as multiplying by 1/6. Students did say those things but more in a vague-only-the-teacher-picks-up-on-it kind of way, but I think if we focused their thinking, it may come out more.

The ending (If you are still reading at this point, that is dedication)….

My Class: Through our group planning, we decided to end with a journal entry/exit ticket asking students to think about where they saw multiplication in the work they did. We thought bringing their attention back to their work, having multiplication in mind, would push students to think about many ideas such as taking a part of a part results in a smaller answer, the order of the fractions resulted in the same answer and the denominators multiply because you have to think in terms of the whole. So…I did not really get the results we were looking for. Don’t get me wrong, I had some wonderful responses I used to guide my instruction, however many left me with no idea of what they were thinking about the fraction of a fraction work. This was mainly because of part b in each question asking students about cost. They ALL saw multiplying there…you have 5 pieces, each is $1 so 5 x $1 = $5. That really didn’t help me as a piece of formative assessment work. Here are some of the ones more along the line of matching our goal that we used in adjusting for Alicia’s lesson: (sorry about the lightness, my copier at school obviously has some issues)

9  14131110  12

Alicia’s Class: After reviewing my student work, we decided for Alicia to use a ticket that had a new problem, 3/5 of a pan 1/2 full, followed by the question about multiplication noticings based only on the first parts of each question and the ticket. MUCH better read of what the students understood. Even if they could not see multiplication, students were able to represent the problem and come to a solution. It was more informative in terms of where to start building with that students and where they were in terms of our mathematical goal.  It was interesting to find, in reviewing her tickets, that students she and Jennie thought “had it” during the lesson could not do it independently on the exit pass. We went back to their work for the day and they did seem to have the correct work and responses, however how much of that was table work?

That got me thinking how it would be nice to know what student writing was done during individual think time before the group work started. I think next group task, I will have different-colored colored pencils on the table…one color for individual time and then switch colors for group work. Then I may be able to better see a student’s thinking.

My Reflection on the Process: Throughout this entire process, I found myself saying “we did…” and “our lesson…” A LOT. It became not just my lesson that I was planning for my students, or a lesson I was observing to give feedback, but instead a wonderful collaboration in which the entire goal was to make it the best possible learning experience for the students. In planning the initial lesson, there were things that didn’t go as we thought, things we hadn’t anticipated would happen, and connections that we thought would clearly come about, that didn’t. The fantastic part is, we got a Take 2. We had the chance to talk through why things didn’t work and how we can improve for the next take. After a lesson, I am typically left wondering, “What if I had done…,” and although it was not taught again with my students, I got to see that “What ifs” play out…and it was so much better!

I am a HUGE fan of coaching to improve teacher practice and this type of lesson study took it to a whole new level for me. How wonderful would it be to see the lesson I am going to teach play out before I really have to teach it? I am really thinking hard about how this could work in a school or district. The platform of Teaching Channel Teams was invaluable in this process and I think would be an integral component in making this work….

The End….well, of just this piece….MUCH more to come,

Kristin