Making Sense of Decimals in 5th Grade, Pt.2

So about that number line I mentioned in part 1. I love clothesline math! I loved it before I knew there was a blog about it. I loved it when a colleague said, “Hey, did you see this?” There are so many possibilities with putting numbers on a number line, and discussing things like:

  • Convince me 7/4 is in the correct spot.
  • Why did Johnny put 1.2 on top of 12/10?
  • Do you agree that 1.25 belongs between 1.3 and 1.4? Why or why not?

The discussion with the partners is so interesting. And then putting numbers on your own white board. And the formative assessment that comes with that! And then magically having numbers in the correct order without having to stack them on top of each other with place values lined up.

Quite honestly, I’m not sure how you teach math without a number line at this point. I’ve had so much fun with it that it makes me want to be back in the classroom exploring it. But, I’m enjoying exploring it with my teachers as well.

So, my 5th grade teacher who was looking for strategies probably felt my excitement about doing the number line, so she agreed to try it. After warming up with some choral counting, we jumped right in. I had consulted Kristin Gray’s blog, KG Math Minds, and got some great ideas there. Check the post out here to get far better description of how I planned using many of her ideas. A couple of the ideas she reflected on were where to place the 0 and how many cards to give partners. I decided I’d start the conversation by putting the 0 in the middle of the number line. I asked the students if that was OK, and why. They agreed that the left was negative and the right was positive. Since that seemed clear, I moved the 0 to the left and told them we’d only be focusing on positive numbers today. Then I held up 1/2. Where should this go, I asked. The students were hesitant. Rightfully so, as there was no benchmark for the whole. A brave soul came up and placed it. We had a brief conversation about the choice, and the realization dawned that it didn’t really matter till we placed the 1. So, I tested the theory. Instead of a 1, I held up a 2. A brave volunteer came to place it. She placed it so that it the 1/2 was equidistant from the 0 and the 2.

Aha! Misconception. Keeping my poker face (isn’t that hard when you want to raise an eyebrow and say nope, try again?), I applauded her bravery and asked the class, how do we feel about the placement? Talk to your partner about why you agree or disagree. The students worked to find the words to say that 1/2 should be closer to 0 than 2. They eventually got there.

Now the cards were spread out, and we added in the 1 to help settle some disagreements. So, I handed out the decimal cards to pairs of students. Here is the end result:

First, don’t make fun. I know the cards are too small. I was anticipating bringing them to the carpet for a closer experience, and I thought my number line would be much smaller. I hated the overlap when I used full sized index cards in the past and was trying to avoid that. Plus, proper planning prevents poor performance – I should have planned this part out with the teacher but the time crunch got to me. So, I overthought this a bit and ended up with a problem. But, the kids assured me they could see it, and that my phone did not take an accurate picture (side note…why do iPhone pics look so much further away than the reality?)

Second, I took the picture at the very end when I realized I had no visuals to share with others about the clothesline. It didn’t start as well spaced out. Plus, there were some answers that needed more immediate revision, and we had to discuss those. Coaching note…take more pictures during lessons. Visualization is important!

One of my favorite events of the lesson was the partnership that had 7/4. During the questioning in choral counting and the beginning of the number line, I had noticed a couple partnerships talking at a pretty high level. So as I passed out the decimal cards, I handed out a couple of fractions that I figured would be a little more challenging at this stage in their learning. This partnership put 7/4 to the right of 2…suggesting it was more than 2. I let that go for a bit, as we debated a few of the other decimals (mainly spacing them accurately between 1 and 2.) Suddenly one of the partner’s hand went up. When I called on him, he asked if he could move the fraction. I agreed and he moved it to the left of 2. Why did he move it? Ensuing partner talk.

During the partner talk I chatted for a bit with the teacher. She was surprised at some of the discussions her kids were having, but also at some of the misconceptions that surfaced in the placement of the numbers. She realized that they needed to do some more work with placing numbers accurately on the number line.

In the debrief later on, the teacher shared with me that she had continued the number line work, and was using it as they considered comparison of decimals. She was interested in using it as they moved into fractions. I’m excited for her as she journeys on with the number line, and I’m excited for her students who begged me to stay and keep working with the number line. I’m excited for the student who discovered his own error and was willing to revise his work in front of the group.

In my own reflection, my math goal was really to seek information about their understanding of numbers on a number line. The difficult part about demos is that even though a teacher shares knowledge and data on where the children are at, there is still the reality of being in the room with students day in and day out, and the deep understanding that comes with knowing what you’ve taught, where they’ve struggled, and where that zone of achievable challenge is. The teacher and I both saw that moving forward they needed to work on the distance between numbers, justifying that distance, and understanding the link between fractions and decimals. Too often we rely on kids “understanding” as demonstrated by a conversion worksheet, but we discover in the number line that their “understanding” is limited to a procedure. I do wonder how much of this affects 5th grade SBAC scores. I look forward to seeing what they do with the number line over the next few weeks!

Any best practice ideas for implementing the clothesline in the classroom? Comment below!

Making Sense of Decimals in 5th Grade, Pt. 1

A fifth grade teacher shared with me last month her concerns about her students that were struggling with decimals. They didn’t understand place value. They had trouble identifying greater than or less than. They confused the tenths with the hundredths.

One of the ideas she was interested in pursuing was Choral Counting. We had done work in the past on various talk routines, including Number Talks and Which One Doesn’t Belong. I consulted another coach (remember how I get by with a little help from my friends?) and made sure I was clear on the strategy.

So we started simple. A choral count by 0.25. I used this planning tool my coach colleague suggested to think ahead about what I would scribe on the board.

I planned to stop and look at patterns, but I realized that my structure didn’t lend itself as well for beginning choral counters. So, I changed the number of rows.

That made more sense. A pattern of increasing by one in each column was easily distinguishable with four rows. We could look at why that happens (0.25 is equivalent to 1/4, and our numbers are grouped in 4’s.) Hmm…

They picked up on this pretty quickly. All I had to do was ask:

  • What do you notice?
  • Do you notice any patterns?
  • Can you predict what would be ____? (I put that blank space in the 3rd row of the next column).
  • How could you prove it without counting?

The teacher liked this strategy, but still felt like her students were struggling with the idea of greater than, less than, and the concept of decimals as compared to whole numbers. She wanted to explore the clothesline number line.

It was a couple of weeks later, and I wanted to see if the students remembered the choral counting. I remembered that they thought it was pretty easy, and they were engaged with it, so I thought we’d warm up with a choral count. So, we counted by 0.03. This made for some fascinating conversations that went a little longer than I planned. I wrote 0.03, 0.06, 0.09, and then asked:

  • What number comes next? How do you know? – That elicited 1.2 and 0.12. So I wrote them both up there.
  • Convince me that one of these is correct. – They practiced with their partners first.

Once the group was convinced that it was 0.12 (one of the students had said, well, there are 12 hundredths and that is the same as one tenth and 2 hundredths), then we kept going on. Eventually the count looked like this (without the fourth column):

So I asked if there were any patterns. I waited (did I wait 10 seconds? I need to work on that), then gave them partner talk time so they could orally rehearse. Sometimes I pull sticks after this but this time I preselected a volunteer. He can be a reluctant student, but he noticed a pattern of 12’s with his partner, which he explained to the class that at the bottom he sees 12 x 1, 12 x 2, 12 x 3. So I wrote that on our count, purposefully leaving the misconception on the board. I asked, if that is true, what comes next? A student volunteered 48. So I wrote that on the board, then asked the students, what do we think? Does that make sense?

Discussion ensued, and I noticed it got more lively as they started to realize what happened. I listened in to conversations, and once I was comfortable that the partnerships were prepared, I pulled a stick. The student explained that it should be 0.48. I asked, “Why?” Another student explained the pattern was counting by 12 hundredths, not twelves. So, I changed 48 to 0.48. I again asked if it made sense. They looked around at each other. This time I remembered my wait time! Eventually the partner talk picked up, I called a volunteer, and the student explained that the 12 x 1 and so on should have been 0.12 and so on. Aha! We spoke for a minute about the importance of precision in math (SMP 6), and then moved on.

How long did this routine take? About 10 minutes. Lots of partner talk and making sense of math, with a very easy to implement routine. In part 2 I’ll talk about the number line.

I Get By With a Little Help From My Friends

I cringe just a bit when I hear coaching referred to as a lonely job. The job requires nearly constant collaboration, and so I’m talking with someone most of the day. Really, teaching can be a lonely job too. You spend the day with your students, none of whom can collaborate with you on the professional level, and at the end of the day, you’re faced with all kinds of things to do.

This is where I feel like the power of the coach comes in. No one wants to be wrong or treated as though they were being fixed. But many of us want someone to talk to who will be focused on our problems. A teacher colleague next door might be helpful, but s/he has their own problems that may take over the conversation. A coach comes in and is focused on you and your needs. It’s one of my favorite parts of the job, and has helped me overcome some guilt over leaving the classroom. I spent my first six months on the job haunted by this meme as I pondered my role as a coach:

Now I know better. I spent time yesterday talking with a teacher who is investigating guided reading. Now there is an overwhelming task to take on without formal training (yet…it is coming soon…). So, we worked on breaking it down. Why do guided reading? What are you hoping your students will get out of it? What information do you really need to make that happen? How do you get started? There are so many birdwalks to take in this area that talking it out with someone who can paraphrase your thinking, and help you sort through your thoughts, can be exactly what you need. Sure enough, the five minute planned conversation about a running record procedure turned into a 45 minute coaching conversation, and at the end the teacher said, “sometimes it helps just to talk it out.” Exactly!

So who coaches the coaches? It would be unfair to turn around on those teachers we coach and expect them to reciprocate – that’s not their job and they haven’t been trained to do that. They already have more than enough on their plates. Fellow coaches and I have organized a job alike afternoon once a month, where we join together and share out celebrations, ideas, new learning, and challenges. I’ve come across several articles on the power of these coaching get togethers. I was first inspired by the idea of the Mastermind group from the Cult of Pedagogy. But, when talking to other coaches, we thought we might keep it more loose at first – no specific agenda. No one in our group had done this before, and we didn’t want to formalize too much at first. I suppose this is the “forming” part of our group, but I’m not sure that it fully fits that framework, as we don’t have a “performing” goal at this point.

We looked at some coaching models in our first meeting. Although at this point our district has focused the coaching training on Cognitive Coaching, we talked a bit about other models that are out there. Student Centered Coaching is one that we started to investigate briefly, based on an article one member of our group shared.

But the best part was how the group organically shifted into sharing challenges while staying focused on solutions. Venting is a part of any job I suppose, but it doesn’t solve problems. TeachBoost says,

Coaches must keep in mind that they need to restore themselves first before their work of restoring others can happen.

In a job alike group, we can certainly do some venting, and empathize with the challenges each other faces, but we also know we can brainstorm solutions. We didn’t formally do a Problem of Practice protocol, but we did end up investigating one of the challenges in more depth. In the process, I came up with my own new ideas for our guided reading book room while taking into account the pitfalls described by my colleague.

It was exactly what I needed, when I needed it. A reminder that we are not alone in this work. That no one has a perfect gig, nor does anyone expect perfection. No one “fixed” a coach, nor did anyone need “fixing.” Sometimes we just need to talk it out. In the process, we are validated, we are understood, and we feel like problems are not insurmountable.

How do you stay connected with colleagues and avoid the feeling of isolation?

2019 Goals: Writing

In this last of my 2019 goals series, I’m focusing on writing. Hence the blog! I figure if I’m going to help teachers and students write, then I need to be in the process myself. There is actually a record of me saying I hate teaching writing on Facebook, back in 2009 when posting answers to 25 questions about yourself was trending. Shortly thereafter, I came to discover that I actually loved teaching writing. It was through a professional development series that was actually based in technology that I grew to love the writing process. It’s funny how things can work out that way. Take something I love (technology) and then embed something I’m not interested in (writing) and a spark ignites. I’d like to think that same thing happens for our students as well.

So how did I go from hating writing to loving writing? The process was based in helping my students navigate their brand new laptops (our grade level had been given a laptop cart to share). While they were writing letters to pen pals, I started to realize that I couldn’t try to correct everything possible in their writing. It would sound unnatural – not like a 4th grader. I also learned that there would be many, many questions in this process, and I couldn’t spend 10 minutes with each child. Writing conferences became more like writing drive through’s, where I would know of one or two items in the student’s writing that s/he was working on, and I would focus my brief attention there. I never focused on spelling during these writing stages – I’d file that information away for a lesson to be done shortly thereafter. Plus, I made sure they were aware of how to use spell check! The whole world of writing opened up to me – I realized the power of a mini lesson and the power of meeting students where they were at. Now, Google docs (through the use of Google Classroom) really makes the writing process a fun one for teachers and students. Click here for a short overview of what I love about Google Docs.

I’m super excited to take a group of teachers to see Carol Jago at the end of January, as she presents on opinion writing. I love that she is not only talking about the process, but also how to provide feedback and manage the paper load that teaching writing creates.

One of my biggest interests is in the area of mathematical writing, and will likely be the focus of many future blog posts. Currently, I’m working with many teachers on a set of lessons based out of Think It, Show It Mathematics.

This was suggested to me after attending a workshop from a professor at UC Davis, where she had shared lessons they had created from this book. So, we have worked with them and tweaked them for the needs of our teachers. At this point, we’ve had rave reviews on our PD. Teachers who have really focused on mathematical writing are finding that by having a rubric by which to evaluate the work, along with language for the writing itself, has helped their students’ explanations soar. Here are a couple of examples of fourth graders’ recent math writing. Note the use of transition words and a clear statement of the answer.

Clear statement of the answer. What might be your next steps with this student?
What understandings does this student have?

I just happened to grab a couple of snapshots as I was in the room for another purpose – to observe their collaboration as partners! But, I was so struck by the fact that after collaborating with partners, there weren’t any questions of “How do I explain this?” It was fascinating to not only observe their conversations, but then see them explain their thought process so clearly on paper.

I’m looking forward to doing even more investigation into mathematical writing with my teachers, through a few different lenses. First, there’s work from UConn including the original recommendations from the Elementary Mathematical Writing Task Force. Linked there, but I think also important to link here is the article, “Why Should Students Write in Math Class?” from Educational Leadership (2017). Last, is a piece that I found on Twitter a week ago called 17 Prompts For Writing in Mathematics. See a glimpse below:

So that’s it – my three big goals this year are to investigate Intervention, Listening & Speaking, and Writing. Keep in mind though – all of these investigations will also come through the lens of coaching and support, so I certainly intend to blog about coaching as well. What are your goals this year?

2019 Goals – Speaking & Listening

Another goal for 2019 is to locate and develop resources strategies around our Speaking and Listening standards. This is urgent for three main reasons. First and foremost, we know that speaking and listening are key literacy and life pieces. To produce the citizens of tomorrow, it is imperative that we teach them to be effective speakers and active listeners. In looking at the 2020 Top 10 Job Skills prediction, it would seem that consistent instruction and application of listening and speaking would greatly benefit our students.

Second, our new report card includes a grade area for listening and speaking. So, our teachers are actively seeking support and professional development to help them assess these standards.

Last, our test scores indicate that there is a need for better understanding about the listening standards. While I feel the need to make it clear – summative test scores are a snapshot and absolutely do not solely define the child – I do think they are valuable to use in making programmatic decisions. It would be unfortunate for our students to have low scores simply because we didn’t analyze the situation and identify causal factors.

As a result, a group of coaches and I are planning an optional, introductory PD to these standards. Although the idea of listening and speaking is not new – consider our EL supports and PD over the past several years – we recognize the need give sharper focus to these standards. After a solid brainstorming session today we realize that even two hours doesn’t seem like enough. How do we include information about SBAC, ELPAC, the CCSS, prior ELD strategies that could be brought back, resources that are available, resources that we have created, our new report card, and of course the classroom management challenges that speaking and listening can bring?

We are particularly inspired by three Teaching Channel videos:

  • Listening & Speaking: Formative Assessment – in this video, the teacher uses a checklist during an academic conversation in order to record who is meeting the learning targets of the conversation. Simple, low prep activity with high engagement, that is rooted in the standards.
  • Evidence Based Academic Discussion – this is the “prequel” to the prior video, showing how the teacher set up the conversation so that students came to the circle prepared for discussion. In the “exit ticket” there is space at the bottom for students to reflect after the conversation – a great way to look for evidence of listening (since a look on the face can be deceiving!)
  • Formative Assessment: Collaborative Discussions – we were hooked in this video by the use of the poster to help students organize their thinking during their discussions.

We continue to look for formative assessment ideas around the speaking and listening standards that are easy to use and provide good feedback for next steps in instruction. Any suggestions, please leave them in the comments!