Making Number Talks Matter – Book Study!

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I honestly cannot talk about the teaching and learning in my classroom, or across our school, without highlighting Number Talks. I cannot recall the very first time I started Number Talks, but now I cannot imagine my math class without them. My Number Talk journey began with Sherry Parrish’s book and had continued to grow through reading Cathy Humphreys’ and Ruth Parker’s most recent book, Making Number Talks Matter. I implement Number Talks on a regular basis, reflect through writing blog post after blog post and have presented at both NCSM and NCTM around Number Talks. I simply cannot say enough wonderful things about them!

For this reason, it was not surprising when I had the opportunity to meet my fellow Teaching Channel Laureates this summer, that Crystal (@themathdancer) and I struck up an immediate conversation about Number Talks when we began chatting instruction. As an elementary school teacher, I often have middle and high school teachers ask what Number Talks could look like in the secondary classrooms. Would the setting be the same? What would example problems or strings look like? How does the content focus change? How do we get students at that age to engage in these mathematical conversations?…etc. So many questions that I still am trying to wrap my head around.

Fortunately for me, Crystal is a middle school teacher using Number Talks, so our conversations gave me great insight as to how they look and feel in the middle school. We talked extensively about the Number Talk course she had taken with Ruth Parker and shared students’ conversations during Number Talks in our classrooms. It was so exciting to see the connections between our students’ experiences and the path Number Talks can take after my students leave me and enter the middle school.  While I would love the opportunity to plan and observe Crystal’s classroom, and her to visit mine, she lives in Washington state while am on the east coast in Delaware, so our opportunity to collaborate face to face is not a reality.

Enter the wonderful world of technology, amazing resources available through The Teaching Channel, incredible teachers around the world wanting to learn and grow every day, and Ruth and Cathy’s new book Making Number Talks Matter!  Now, Crystal and I do not have to be on this amazing journey alone, but instead we have the opportunity to create an experience in which other educators everywhere can join us!

Together, we have planned and structured a book study unlike any other! Flexible to fit your needs, full of valuable resources, and completely FREE!

Beginning October 5th, each week will be dedicated to one chapter of the book. Conversations will happen on Twitter and Facebook, videos will be posted in our Teaching Channel Teams group, and we will even have a guest appearance by Ruth Parker herself! It is guaranteed to be an amazing learning experience that can only get better with your participation!  

For more information and to register for the book study, follow the links below:

We look forward to learning with you and Getting Better Together!

Check out Crystal’s Number Talks journey here!

-Kristin

Establishing a Culture of Learning …The First Hour

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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.

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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.

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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!

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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…

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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.

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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!

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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

The Clothesline Number Line

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The other day Andrew tweeted this post about the use of the clothesline in math class…

 “Clothesline is the master number sense maker.” says @timsmccaffrey according to @MathProjects. Good call Tim! pic.twitter.com/bVvvJz7CtX

— Andrew Stadel (@mr_stadel) August 19, 2015

I love the clothesline activity and have used it with students in the context of fractions, decimals and percents. When I first started using it, it would be up with pre-marked benchmarks on it and as we talked about relationships and equivalencies we would add them in relation to the appropriate benchmarks. For example, 0 and 1/2 would be already up on the line and in class the students realize that 1/4 is half of that distance, we would put the 1/4 on the line between the pre-marked 0 and 1/2. It was always a work in progress and it was a nice visual, but static.

After designing a PD with Jody Guarino, I realized the power in having the pieces be constantly movable. So we started with an empty number line and students drew fraction cards to put on the line one at a time. Now, there are two ways to launch this… either the students know the range of the cards they are using or they do not. I prefer the do not, but I see benefits and places within grade levels/units that each would be appropriate.

Let’s say the students did not know the range of the fraction cards (and they can have whole numbers mixed in there too) they were pulling. The first student is almost a blind guess because they do not know where this line starts or ends, but from there, there is constant adjustment to do.  It is a bit hard to describe in words so I hope this makes sense…

Student 1 draws 1/2 and places it in the middle of the clothesline assuming that possibly the line goes from 0-1. We ask why they placed it where they did and hope to hear something like “I think the line goes from 0 to 1 so 1/2 is halfway between those.”

Student 2 draws a 2, places it at the end, but then has to adjust where the 1/2 is placed because now 1 must be in the middle of the clothesline. Makes the necessary moves. We of course as why they adjusted the way they did and how they determined how close to place the fractions to the other cards.

Student 3 draws 3/2, no adjustments are necessary at this point, but the student must estimate where 1 would be and do half of the distance to 2 and place the card OR take the 1/2 distance and replicate that three times. Either is awesome. Of course, we ask all kinds of questions about their placements.

…and play continues, but if a 3 is drawn, then things shift tremendously…the fun part is students don’t know so they cannot have their spot pre-planned and everything is adjustable!

After Andrew’s post, I started thinking about two things…what could this look like with whole numbers and what are the differences between starting with a set on the line to move around vs drawing cards and placing them where they go?

I tried to sketch out a few moves and think about what students could be thinking about as they went. I am sure I have not captured all of the possibilities, so I would love to hear ideas in the comments!

First sketch, numbers 0-6 placed randomly on the line to start (top line)

IMG_0499So, we could be looking for students to place the beginning and end (0 and 6) first and then I would be curious to see if they placed the middle number before dealing with the other numbers (line 2). Or, would students order left to right, least to greatest (line 3)? Would be interested how they think about spacing the cards in this situation, do they touch? Is there the same distance between each? What would they think about if we gave them a card with 10 on it after they finished arranging them on the line?

Now, what if we started with just a zero at the left and didn’t tell students what range of numbers we had?

The first card they could draw would be a 2, they place it in the middle. We asked why they placed it there and maybe hope to hear something like “I think the line goes to 4, so 2 is the middle of 0 and 4.” (Line 1)

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Then, let’s say the next student draws a 6. They place it at the end, again assuming it is the highest number and then has to adjust the 2. We ask what their thinking was and hope to hear something like “Since 6 is the end now, 3 must be in the middle and 2 is less than 3.” There could also be some questions around how close the 2 is placed to the middle versus toward the 0. Cool stuff. (line 2).

Next student draws a 10, now the 6 shifts left, just right of the halfway point though because that would be 5. Then the 2 has to shift because the 6 did. Now, interesting questions about all of those moves come out..How did you decide where to place 6? Why did that affect the 2? How does the distance to the 2 relate to the distance to where the 5 would be? to where the 6 is?

Andrew played around with the clothesline with his son here. Great stuff and so many ways to think about how we could use this in our classrooms across all grade levels!

Thanks for the inspiration to blog this Tim, Chris and Andrew!

-Kristin

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A Teacher & Mathematician Mash Up

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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!

My Questions Around Professional Development w/Video

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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

Math Reasoning Stages

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Tracy Zager (@TracyZager) asked for some thoughts/pushback on these stages of math reasoning imagined as a flow, so here are my thoughts based on my experience in the classroom…

Pattern Sniffing –  After students see a pattern I find they continue, using that pattern, for a while before thinking about a generalization. So, maybe “Extending Pattern Using the Pattern” comes after this in my mind?

Wondering – When they wonder, they definitely look at relationships, but I am not sure they wonder if it will always be true at this point? Now that I just wrote that, I am thinking maybe “Extending Pattern Using the Pattern” comes after this one?

Articulating – “Can I communicate what I am seeing happening in a precise way?” I don’t know if they are thinking too much about it at this point but more seeing it happening? Could

I don’t know where this fits necessarily, if it is embedded in one of them, or if it really fits at all:), but there is a point where mathematically students prove a generalization works with certain number and not others because “the numbers have to work that way” (structure) without the conceptual proof of why that is. For example, “Even dimensions of a rectangle will give you an even area.” Students can make the statement that it has to work every time because when you multiply even numbers it is always an even product….true, but isn’t there proof to that. So, it is like a string of proof by depth?

Investigating and Explaining Why – I feel the relationships and patterns question to themselves comes back up here too.

I love thinking about this process for students and the teacher implications between each step. What questions and/or feedback do we give as students go through this that isn’t too helpful or leading, but not too vague that leave them in one spot spinning wheels? Paging @MPershan…

*Chatting with Tracy after I wrote this, she was focused on mathematicians, not students. I find some holds true in both cases to different sophistications. 

Hope that helps a bit Tracy! Hopping on a plane but as soon as I have wifi I will add a couple more question I have to think about around this!

-Kristin

Number Talks vs Number Strings

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When I first saw this image, I have to admit, it didn’t match what I had been calling a Number Talk in my classroom. Having started my journey with Number Talks with Parrish’s book, I saw it as a string of problems with a specific strategy as the focus of the string that got progressively more difficult (which sounds more like a string in reading this slide). As I started creating my own and making variations to them over the past couple of years, I have simply started calling them Math Routines….it was just easier than trying to make things fit into a certain framework in my brain. After going back and forth about terminology, I started looking at these structures more in the sense of purpose than the name and I am finding it very interesting. Please keep in mind as you read, this is completely from my experience with Math Routines in the classroom and I find all of these talks so incredibly valuable!

First comparison: Single problem vs string of problems. In my experience, I think these two designs have a substantial difference in purpose. When I give one problem, I am going for one of two things: a variety of strategies to see where students are in their thinking OR connections/comparisons of multiple strategies. Personally, I like the variety of strategies before and after we have investigated different ideas that could impact their thinking. For both the students and myself, it shows growth and I can use what I find to help me in planning future routines. Connecting and comparing takes a bit longer and, for me, I don’t get as many strategies out because we focus on connecting and comparing only a few for time’s sake.

Second comparison: Difference in having a representation and context. I don’t give a representation or context unless a student brings one up in a explanation. If it comes from them, I go with it, if it doesn’t, I don’t write it up there. If there is a specific representation I am hoping comes up because we have been doing work with it during our math class, but doesn’t, I do have them do a quick journal response. I will ask them something such as, “How was our number talk similar to or different from our work in math class this week?” Then I can read their journals and have some students share the next day about the representation connection they made. I also have had students create contexts after we finish a number talk with a prompt such as, “Write a context that you think fits one of the problems in the string. How does the context change for another problem in the string?” For example if the string has “16 x 4” followed by “16 x 8” I am hoping to see connections between the two problems in the context.

Nothing to compare in the final piece, multiple strategies discussed in both!

In the end, students are talking math with a mathematical purpose so they are win/wins in my book, but I am curious to hear what others think around the purposes of different formats of these routines! Would love to hear other ideas so comment away!

-Kristin

#IntentTalk Chapter 1

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Since it was a bit too much for me to continually tweet, I decided to do a quick blog at lunch!

Principle 1: Discussions Should Achieve a Mathematical Goal

The first week of school my mathematical goals revolve around discussions about students’ mindset in terms of math, as well as the mathematical practices. I found this year that Talking Points and one of the tasks I found on Fawn’s blog fostered those goals. I blogged about it here. My tables are all arranged in groups and the students know from the very first day that discussions will be a huge part of our work together.

Principle 2: Students Need to Know What and How to Share

To support this in my classroom at the start of the school year, I have the students agree upon our class norms. They originate after doing a Number Talk together and reflecting on what we expect as a group during our discussions. I reference these norms throughout the course of the school year.

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Principle 3: Teachers Need to Orient Students to One Another and the Mathematical Ideas

I find a lot this happens during our Number Talks and in then daily in our journal reflections. This is such a focus on my planning of questioning. Asking things such as, “Can you re-explain their thinking in your own word?” or “Did something ____ say change your mind about that way you were thinking?”

Principle 4: Teachers Must Communicate That All Students Are Sense Maker and That Their Ideas are Valued

I think this principle emerges during our very first round of Talking Points of the year when the students my go around their circle with no commenting from others. It allows students the opportunity to speak their ideas without judgement or comment. Being able to change their response on the second round, lets the group know that as we make sense of problems and listen to others, we change our mind, just as we do when we make errors. The freedom I see in their journal entries also reinforces the idea that I value their thinking and know that there is reasoning behind everything they write and do in my class.

“Talk is an important way to build that sense of community and to help children grapple with important mathematical ideas.”

-Kristin

Last Day of Math Class :(

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Today was the last official day I had my students for math. It was a bit sad for me and it was nice to hear some of them say it was “kinda sad” for them too. In moving into a K-5 Math Specialist position next year, I know it will not be the same experience watching a group of students grow over the course of the school year.  It will be great in different ways, but I am really appreciating all of the amazing work my students have done this year.

So…what to do on the last math day after they just had field day yesterday followed by our PBS bowling field trip tomorrow? It is a tough planning!

I first had them look through their two math journals, one from the first half of the year and one from the second half. As their last writing piece, I asked them to write things they noticed in their work over the course of the year after looking through their journal. I only had time to grab one journal today because the end of the year craziness is kicking in, but I plan on following up with a more detailed post later. This one was so powerful and truly gave me goosebumps….

IMG_0955_2After they finished that, I asked them to revisit some of the claims they had written over the course of the year and see if they still thought they were true and could be proven or were not true and needed to be revised. This student had written a claim that when you are multiplying fractions, you could multiply the numerator and denominator to get your answer. As he was proving it just worked for multiplication, he stumbled upon the realization that it worked for division as well. He then worked through a few more division problems and it was such an amazing explanation!

He revised his claim…

IMG_0956_2IMG_0957_2I promise to follow up with some really amazing work they did on the last day when summer is here and there is a chance to breathe 🙂

-Kristin

Growth Patter Number Talk….3rd Times a Charm

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Over the past couple of days, with my homeroom, I have tried a few strings of numbers to bring out the different ideas that are important when thinking about growth patterns and finding any term in a sequence, Here and here. Both days brought out many great ideas, conversations, and disagreements, however I couldn’t help but feeling the ideas we talked about in two days, could have been achieved in one and felt a little more connected. I knew it was completely the way I posed the problems, so when my second class came in yesterday, after missing a couple days of math due to testing, I was excited to try and adjust my previous work.

Apologize for the messy board, but I still cannot seem to get a handle on that recording thing…

I started with having a student count by 6’s and wrote that in blue. I stopped them at 4 because I was asking about the 10th term and wanted to see if some would figure our 5th and double. I think that is an interesting thing to think about when the start is different so I wanted it up there. I asked 10th term? 60. Ways to get there? 6 x 10 and, unlike my prediction of doubling 30, one student said 24 x 2 because two group of 4 of them is 8 and then two more 6’s (12) is added to that to make 60. I asked 100th term? 600. 2,000th term? 12,000. I asked them how they were getting those without counting and I got “I did 6 times 100” “I did 6 x 2000” and then one student said you could do any number by multiplying it by 6. I asked how he wanted to write that and I wrote that in green. Another student, who has done Visual Patterns with me in our RTI group, said, “We can also write that as 6 times n = Answer.” I asked them to turn and talk to a neighbor if they thought that meant the same thing. We had all yeses and I had some student prove it. I did the same thing with 8’s and wrote that in orange. They started using “A” for “Answer.”

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After that, I asked them to to count by 3’s starting with 6 and stopped them at 15… Asked for the 10th term and got, as expected, 30 and 33. Then the conversation took off with proofs and some really important ideas that was hoping would emerge. I love it when the class is practically divided in half on an answer, we had the 30’s and the 33’s. I asked a 30 to explain how he got the answer and he quickly said 3 x 10=30. I saw a lot of agreement, so I asked for a 33 to share their reasoning. A student said that we “need the beginning number, three, to find out where the tenth one is. 3 x 10 is 30 but then you started three ahead of that so you add 3 to 30.” I wrote that down on the board.

A student then said something that made me have a realization, “It shouldn’t change because you are still doing 10 jumps of 3, so it HAS to be 30. 33 is 3 x 11.” In my last class I had a student who kept insisting that the 10th term remain the same no matter where we started and I could not figure out what they were trying to articulate. NOW, I understand. 30 will always be the distance between wherever we start in the sequence and the 10th term, but not the tenth from the true beginning. AH HA!

IMG_0774_2So, the beginning number was suddenly becoming very important and articulating “10th term from where” was having students agreeing that the 10th term starting from the 6 was going to be 33 but when thinking about a rule for the pattern we needed the true beginning. We were just about to head back to our desks to continue our work when a student (different than the one who had originally said it) said that we could write this one “3 x n + 3 = A” because you have to “add the three you are missing from the beginning to get the answer.” I had them turn, talk and try a few terms out and see what they thought. It was all wrapping up nicely (I was excited about it) when another student said, “You could also write 6 + (3 x n) since you are starting at 6” ….oh goodness, they just don’t ever let it end and I love it:) A disagreement arose that it would have to be “6 + (3 x n -3) because of that extra jump of 3 to start at 6.”

I always hate to say that time got the best of me, but I had missed this group for 2 days of math and I saw this conversation going lonnnnng so I had them write those ideas down in their journal to kick off our class on Monday!

I love when I have the chance to refine ideas that don’t go exactly as I had hoped they would, especially when I know it was completely how I posed the problem or asked the question. After a couple days of talks not connecting as I hoped they would, third time was a charm!

-Kristin