Episode #42: How Do I Spiral My Math Standards Without Spiralling Out Of Control! [Part 2 of A Math Mentoring Moment]

Sep 16, 2019 | Podcast | 2 comments


This week Michael Rubin from our Math Mentoring Moment Episode #40 is back to continue the conversation from proceeding vs. pivoting to organizing his course content. Yes, this episode packs in a great conversation about spiralling your math standards WITHOUT spiralling out of control.

If you haven’t yet listened to episode #40, be sure to STOP – and GO LISTEN NOW! If you already HAVE listened, then let’s pick up the conversation where we left off and pivot the conversation into a deep dive around spiralling Michael’s math course for the very first time. You’ll grab some great value bombs from this episode…

Let’s get ready to talk SPIRALLING your math curriculum!

You’ll Learn

  • What is Spiralling & Why Everyone Should Be Doing It?
  • What Should I Consider When Planning To Spiral?
  • How Long Should A Cycle Typically Take?
  • How Can I Help Struggling Students While Spiralling?
  • How Can I Get Started?


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Michael Rubin: If I’m going to take this task, so I’m going to try to consolidate it. I don’t imagine myself sitting there and saying, “Hey why don’t we try this?” I imagine myself posting a solution that somebody else did and say, “This person or this student or whatever, did this. What did they do? Why did it work? Compare and contrast it to what you did, and have them think of those things, so that even in that consolidation I’m not just talking at them.” So I’ve got that idea of this purposeful practice [crosstalk 00:00:25]-

Kyle Pearce: You’re listening to a snippet of our conversation with Michael Rubin, a Math Moment Maker who is back to join us for a continuation of Episode #40, because the conversation was so rich, we couldn’t leave it at just one episode.

Jon Orr: Yeah. So if you haven’t listened to Episode #40, be sure to stop and go listen right now. If you already have listened, then let’s pick up the conversation where we left off and pivot the conversation into a deep dive around spiraling his math course for the first time. You’ll grab some great value bombs from this episode.

Kyle Pearce: Let’s get ready to talk spiraling your math curriculum. Here we go. Welcome to the Making Math Moments That Matter Podcast. I’m Kyle Pearce from TapIntoTeamMinds.com.

Jon Orr: And I’m Jon Orr from MrOrr-IsAGeek.com. We are two math teachers who together-

Kyle Pearce: With you the community of educators worldwide who want to build and deliver math lessons that spark engagement, [inaudible 00:01:37] learning and ignite teacher action. Welcome to Episode #42. How do I spiral my curriculum for the very first time? This is part two of Math Mentoring Moment episode with Michael Rubin from Northern California. Let’s do this.

Jon Orr: Throughout this conversation with Michael, you’ll be hearing about this dedicated math teacher who is a clearly life-long learner. Not only does Michael continually reflect on his own practice, in order to focus his mathematics professional development on his own educator-learning needs, but he is also extremely intentional in how he organizes his long-range course plans.

Kyle Pearce: One of the ways Michael has continued to fuel his own professional learning, was by participating in our six-module, online workshop. Michael continues to stretch his content knowledge and pedagogical practice by participating in our ongoing Make Math Moments Academy, so he can continue to access the online workshop content, as well as engage in new learning modules, math tasks and the Math Moment Maker Community area.

Jon Orr: Are you ready to do a deep dive with us in the online workshop like Michael did last fall?

Kyle Pearce: Well, you’re in luck because we run this online workshop twice a year, once in the spring and once in the fall. So fall 2019 registration is open now. It just opened on Friday September 13, but it will close on Friday September 27, 2019. So be sure to learn more at MakeMathMoments.com/OnlineWorkshop.

Jon Orr: And if taking a deep dive alone isn’t enough, the workshop is accredited for two to four PD credit hours to Brandman University. If you’re interested in learning more about registering, be sure to check out MakeMathMoments.com/OnlineWorkshop. If you’re listening after the fall 2019 registration, you can still head over to MakeMathMoments.com/OnlineWorkshop to join the waiting list, in order to get notified of your next opportunity to participate.

Kyle Pearce: That is MakeMathMoments.com/OnlineWorkshop.

Jon Orr: And now here is the second half of our chat with Michael.

Kyle Pearce: Welcome back on the podcast Michael. We had you on just a few episodes ago in Episode 40, so welcome back. How have you been seen we last chatted?

Michael Rubin: I’ve been great and thank you for having me back. I’ve been super busy. My wife runs a summer camp, so is taking care of a couple of 10-year-old girls for a few weeks, and got everybody at the camp mad at me because I taught them Nim. And they went and they just kept beating everybody, once they figured it out, so that was a lot of fun. And I’ve been working on spiraling my curriculum so [crosstalk 00:04:28]-

Kyle Pearce: Oh beautiful. Beautiful. Do you mind just quick reminder for those who may have either not been with us on Episode 40 or they haven’t heard it yet, who are you? Where are you coming from? What’s your coaching role or teaching role, I should say? And then we’ll dive right into the meat and potatoes here.

Michael Rubin: Yeah. So my name’s Michael Rubin. I’ve been teacher on special assignment in Northern California in The Bay Area. I’m going back to the classroom next year. I’ll mostly be teaching Course 1 which is an integrated freshman math to English language learners, and either a Course 3 or precalc.

Kyle Pearce: Nice. Nice.

Jon Orr: So when we had you back Michael, or had you back on Episode 40, you were really concerned and really diving into this idea of spiraling your math curriculum. And you had questions and struggles, and we realized that there was so much about spiraling curriculum, that we had to extend it to two episodes. So that’s why we have you back and I think we just want to dive right into questions you have about spiraling. So if you’re listening to this right now, you haven’t listened to Episode 40, you might want to start there because we talk about what spiraling is and then we move into talking about some struggles on spiraling.

Kyle Pearce: Yeah. And actually as we get and dive in here, John and I are actually on a video call with Michael on this episode. So what we’re going to be doing for our academy members, is on and off we’re going to actually be doing some screen sharing. I’ll do it from probably my end, probably easiest.

Kyle Pearce: But just so those who are listening at home, we are going to be referencing some of the work that Michael’s been doing around spiraling and really just providing one perspective of what this might look like for this particular course that Michael’s working on. So Michael, do you mind framing that out for us a little bit? What got you on spiraling? What’s your intent here of actually trying to spiral a curriculum and then we’ll start diving in and figuring out where you started and where you’re heading to with that plan?

Michael Rubin: I was first introduced to the idea of spiraling in your grassroots workshop. I’d never heard of it before. It had never been in my frame of reference, so I was eagerly anticipating it as the modules came out every week. I’m like, “What’s this thing called spiraling? Is this a task? Are they going to be drawing constructions?” I didn’t know what it was.

Michael Rubin: And when I saw it I really liked it because one of the struggles that we deal with as a school and as a district, is that we get a lot of incoming students throughout the year. And what’s very difficult with that is let’s say you teach Unit 1 on solving equations in Course 1. And then you’re done with solving equations and then the kids come in March, so they don’t get any of that.

Michael Rubin: And so the main reason that I wanted to do it other than the other benefits that you talked about, just to give kids an opportunity to engage with these concepts, even if they do come in sometime in the middle of the year.

Kyle Pearce: So that’s great to hear. There’s some intentionality behind that. This idea of helping students with retention is key. I know for me, one of the big pieces and I don’t know if I knew it was going to happen the first time I did try to spiral, just even this idea of really being able to start at that surface level with a bunch of ideas, rather than going really deep, really fast and losing someone along the way. That for me was really, really big.

Kyle Pearce: I’m wondering, so you’re framing out this course and did you have a starting point? I’m going to start sharing on the screen, for those in the academy who are watching this little mini course here but did you have a starting point? What got you thinking along these lines, because I’m picturing myself sitting at home listening to this or watching this and saying, “Okay, he’s going to spiral like I have no idea where to begin.” So can you help someone get into your mind as to where you started and how that’s progressing along.

Michael Rubin: Yeah. So I had no idea where to really start. You guys had given a lot of good advice. So I’d figured I’d start with the scope and sequence documents that our district had come up with, and really to go through those because they were written in terms of units, to go through them, find the big ideas and come up with the main topics, which I found were algebra, geometry and statistics, the main subject areas, and then break those up into a limited number of primary objectives.

Michael Rubin: And one of the reasons that I want to do that is, and this is beyond the scope of spiraling I think, but I think it works with it as well, is that I do plan on switching to standards base grading, which we don’t need to get into. But I was like, “I don’t want to have 90 different objectives or whatever, so I’d like to do 13 that we can work on with those little sub-objectives over time.”

Michael Rubin: So once I came up with those 13 primary objectives, I broke them up into sub-objectives and then I listened to one of your podcasts and you introduced me to this No, Understand, Do Form, so I went through and fixed them that way. That actually helped a lot with the spiraling because I was able to start thinking about well, when I spiral and I’m spiraling by what kids can do like the skills, am I spiraling based on the understandings like how am I going to do that? So that helped me figure that out a little bit more.

Michael Rubin: And so once I did that, I didn’t really know what to do so I was like, “You know, I’m just going to start searching for tasks.” And I had a bunch of websites, yours, TapIntoTeamMinds.com, Jon’s, mrorr-isageek.com. I had the spreadsheets from Andrew Stabell, Dan Myer. I also had some websites I’d collected from, I’m not sure how to say it, Fawn Newman I think is how you say it?

Kyle Pearce: [Fawn Nguyen 00:10:03].

Michael Rubin: Fawn Nguyen. Yeah. All the Twitter peeps and if listeners aren’t on Twitter, I highly recommend it. Even this morning I was struggling with thinking about tasks for geometry which I’ve had trouble finding. And I think it was John Roe who posted a whole list of geometry tasks and activities and inquiry-based stuff, so it’s really helpful.

Michael Rubin: And so I just started going through tasks and that’s where I left off. I was just about going into tasks, and my general idea the time was that once I got the task I was going to start spiraling by task. Things have changed since then, but that’s an update from last time and where I was and when I left you guys last.

Kyle Pearce: Amazing.

Jon Orr: Okay. So we started that we too. We talked about that before, that we started with tasks and teaching through activity-based tasks instead of just saying, “I’m going to grab 1.1 from the chapter and then I’m going to do 1.2, but then I’m going to switch it up and I’m going to do 2.1 and 2.2 and then I’m going to go 3.”

Jon Orr: We talked last time on the podcast, the benefit is to teach through activities and tasks, rather than just chunk it up, for so many reasons on connecting ideas together and also for retention. And I guess I’m interested when, because we did it through tasks like that to begin with and then you said your ideas have changed since then. What do you mean by that?

Michael Rubin: I’ve never taught through task, so I spent many, many hours going through tasks and trying to think about how I’d teach them. Think about the main concepts that I could get out of them. I know that I was going through a lot of your tasks and you’d purpose them for one topic and I was like, “Oh, but I think I could kind of do this with that.” And I realized if I sat down and tried to plan out a lesson for each one of these, I’d get through at most, 15 by the end of the summer and I’d be no closer to my goal.

Michael Rubin: So I pivoted instead of proceeding as planned, and decided once I’d found a bunch of tasks that really and I almost feel like I think was strategy number seven where you organize by units, instead of looking at the tasks so much and thinking about how I’m going to spiral out the tasks, I decided instead that I would turn my KUD statements into learning progressions and I used your format.

Michael Rubin: You had one a while back, which was all the way from, I think, counting on or something, to linear relations. And I used that to come up with a general progression, like what are the ideas that students should be comfortable with first, as we go through the progression. And then when I spiraled, I spiraled based on the progressions figuring that what I’ll do is I’ll insert the task later.

Michael Rubin: So I don’t think I’m doing exactly what the two of you did, but not being as familiar with the task and what concepts come out, I feel like it’s going to be a lot more sanity inducing for me.

Kyle Pearce: Yeah. I think if I could it again and again, I always look at and think and we can dwell like hindsight’s 20/20, all those things. But knowing what I now know, I think that might be a great way to go about it because now you know you have the intentionality at front and center. And now you can focus on we’ll call, I know backfilling sounds like a sloppy way to say it, but you’re taking now content and tasks and activities and all of these things and you’re starting to place them in.

Kyle Pearce: And some of those as well, it might be really convenient to toss in hey a Dan Meyer task or one of John’s tasks or anything John Roe or any of those things fit here and there, but then also that there’s going to be some little gaps. And then now it’s okay, now that I see this gap it’s like, I can focus my attention on how can I address that piece and ensure that I’m pulling out the learning that I’m hoping that know, understand and do.

Kyle Pearce: And for those in the academy who are watching here, I’ve just been scrolling through this particular sheet, what is your I can statements. At the top they were I can statements, and then you’ve taken those I can statements and it looks like for each concept, I’m just going to flip right to the top here, really quickly here. So you have your overall objectives that you’ve included, so you have a collaboration as an objective that you have. You have reasoning quantitatively, you have reasoning structure, reasoning repetition, algebra, single variable, algebra inequality.

Kyle Pearce: I really like how you’ve started with these big ideas and you’ve tried to figure out, okay, what do I want the kids to walk away with? And then you’ve taken these and you have one I can for each of these, which I think is great because you’ve essentially focused on okay, how am I going to hit that for this one concept?

Kyle Pearce: So for statistics I can summarize, represent and interpret data on a single count or measurement variable. And then now you go, “All right, what does that mean? Now what do I need? What do kids needs to know, understand and do in order for that I can statement to become a reality? So I really like how you started there, and you’re now fleshing those things out. And I think oftentimes what we do is the opposite because of the way our curriculum is written and designed, you’re given all these standards in the US or all these expectations if you’re here in Ontario like Jon and I, and then you’re looking to address every single one of these little tiny pieces, and you’re not even sure how they all interconnect.

Kyle Pearce: And if you go back that Episode #10 with Jo Boaler, she says something along those lines about this idea of when we create curriculum, we’re taking these big ideas, or the curriculum writers are taking these big ideas and they’re chopping them up into a huge number of little, tiny pieces, and then they get scattered and the connections are lost.

Kyle Pearce: So I just want to make sure that we don’t overlook the idea of starting with these big ideas and really going, “Okay, what do I want these kids to walk away from?” Because a lot of the little, tiny pieces may get lost along the way, especially if I’m introducing them in little siloes. I’m introducing this little concept because I have to but not explicitly connecting it to one of these bigger items, so I love that.

Kyle Pearce: I’m wondering, where did these overall big ideas come from for you? What inspired them? Did they come from somewhere or were a mix from different places? Just for those at home so that they know where they might want to start digging in.

Michael Rubin: Yeah. So they were definitely a mix. So I’ve got these 13 objectives, the last 9 are the content objectives. And so I got these big ideas from our district’s scope and sequence document, and so for each unit that the district created, there’s a topic and then it’s got guiding questions, the main objectives and then essential skills. And I went through all of those to figure out what are these big ideas, and then I rearranged them.

Michael Rubin: So some of the units I thought there were more connections between two of them, then I know that the district did use the textbook a lot and trying to figure out how to break things down because a lot of our teachers do use, just teach out of the textbook, and so I wanted to make them a little more connected in some ways.I also didn’t want 30 of them since I’m going to be assessing students on these, just for ease of communication with the kids.

Michael Rubin: And then the collaboration objective I came up with, again referring back to Jo Boaler. She talks a lot about, and I think you guys have talked about this too, assess what you value. That if you’re just assessing the content then you’re not showing them that you value the collaboration or you value the reasoning skills, so I decided to put those in there.

Michael Rubin: The collaboration one I worked on myself, but the Routines For Reasoning, I actually have the book with me this time, by Grace Kelemanik and Amy Lucenta and Susan Janssen Creighton, I pretty much took those reasoning objectives right out of that book. I went through and figured out what are the main ideas in there. They refer specifically to the mathematical practices, the ones that are more thinking oriented like how do mathematicians think, And my plan is use some of their routines and reflect that in some way on the assessments. So I might have kids draw a picture on an assessment to reason quantitatively, so I included those as well. I highly recommend that book [crosstalk 00:18:29].

Jon Orr: Can you say the title again?

Michael Rubin: It’s Routines For Reasoning. It’s called Fostering the Mathematical Practices in All Students. So it’s got a great introduction and it teaches the mathematical practices through routines, specifically to support English language learners and students with learning disabilities.

Jon Orr: Michael, I want to go back. When you were talking with your big ideas on your sheet and you listed them, and then you said you changed the orders of them around based on some connections, I wonder if you can share which ones you decided to group together and why those ones?

Michael Rubin: Yeah. So for example, one of the units in our scope and sequence, is just an entire unit on arithmetic sequences. We just deal with the arithmetic sequences separate from solving linear equations and linear relations, whereas I felt that those were inherently linked. So instead of treating arithmetic sequences as its own thing, I felt like the arithmetic sequences could really drive a lot of our investigations into linear equations and even just solving equations in one variable, and they were super visual, so I think that they’ll help a lot with my students in particular.

Michael Rubin: So that’s an example of where even though I know my colleagues will be spending two weeks or three weeks just on arithmetic sequences, I’m just wrapping that in to everything else. So I won’t specifically be teaching that as a unit, I’m just bringing it up as we go through single variable and algebra and the linear relations.

Jon Orr: That’s a great example of how textbooks or other documents, have chunked things up to keep them separate, but you can save time by linking other things together.

Kyle Pearce: Yeah. I’m looking here. I just scrolled down to the, so that’s your fifth objective. “So algebra can write linear expressions and equations in one variable and use them to solve problems.” And I see you have four knowledge chunks or objectives here, so you have A sub N as the Nth term of a sequence A. The order of operations is something you want them to know. The absolute value of a number is the distance from … And so on and so forth, and then, A linear expression or equation contains a single variable with no exponents or radicals.”

Kyle Pearce: So you’ve got some key pieces for things you want kids to know when they walk out of there, and then you also have some key things for understanding as well. And you can see that you’ve taken these ideas and traditionally they’re separate but you’re bringing them together and it’s you’re sequencing it in a slightly different way than a textbook would. And I feel like you’re providing students with a rich opportunity to make those connections, so that they don’t just see it as some couple day thing, and then you move on and you never come back to it. So super cool. I love it.

Michael Rubin: Yeah. And you might also notice, I put some little, I guess symbols in there, C squared and P squared. I put in some little indicators.

Kyle Pearce: Yeah. I love it, so your C squared it looks like is the common concept throughout.

Michael Rubin: Yeah. Throughout.

Kyle Pearce: P squared use in purposeful practice, so I really like how you thought those things through. What does that mean for you as you laid them out?

Michael Rubin: Yeah. So this was a process I went through before I actually started spiraling the ask, and I went through and I counted all the what can students do, what do I want them to do. And the nine objectives, I had 92 statements. It was like, “Well, I can’t. I’ve got 180 days. I don’t want to break them down and be, ‘Okay, we’re working on that one.'” And I realized that there were bigger ideas or concepts or things you might bring in task, and that there were a lot of others that just either were an extension of something.

Michael Rubin: So I can solve equations for which there is one solution, multiple solutions, no solutions and infinitely many solutions. That might not be something I’m going to find a task for, but that would be a great, I’m thinking of maybe an open middle problem or something, where you give kids and say, “Okay, here’s some sort of prefab equation with some blanks in it. I want you now to fill in the numbers, so that there’s one solution or no solutions or infinitely many solutions.”

Michael Rubin: And it can be something to get kids thinking about it, where it’s not necessarily a task, but it’s a great extension, purposeful practice of students that are understanding, solving equations really well. It’s something to bite their teeth into and practice, but it’s not something I need to go out and try to find a task for.

Kyle Pearce: And I think too, sometimes you can even take that idea of what is a task and maybe for some people, I know I used to always think it has to be a three act math task for everything, but an open middle, I would call that an open-middle task as well. It just might be serving or it might be coming from a different angle, so from the angle of a challenge or whatever it might be, but it could still come under that umbrella of task. So just things for people to think about at home, that we don’t have to have this really flashy task for everything.

Kyle Pearce: Sometimes it’s the challenge and as we build that culture of not answer getting but for actually doing the math and actually challenging ourselves and building productive struggle, or at least this willingness to engage in productive struggle. I think those opportunities for grabbing tasks, as long as there’s something there like that element.

Kyle Pearce: It’s for me, curiosity is always great but challenge is another reason, or just when it’s a puzzle. You give them something that’s more like a puzzle and it’s like you really have to do some deep thinking about it as well. It might not necessarily have the super-flashing context, but don’t feel bad if you want to label that activity as a purposeful task in your classroom.

Michael Rubin: Yeah. You just summarized it. I was taking notes about how I was going to respond as you were saying, and you just addressed everything I was say. So I think the only thing I’ll add to that, just so we’re clear because I haven’t talked about my general plan and I think it might help because I started formulating my general plan as I was going through and doing these. I worked backwards from how much time we have in the school year, and thinking about them not as individual chunks but as a complete spiral.

Michael Rubin: What’s a complete spiral going to look like? And as I was thinking about that I was like, “Okay, let’s say a complete spiral is like two weeks. I can do between 13 and 16 complete spirals depending on time, and that will give me a lot of flexibility. Then as I break those down, what am I going to be doing in those two-week chunks?” And the way I was thinking about tasks was like, those are the introductory ideas.

Michael Rubin: So I’m going to use a task, and a task is something that I’m going to spend maybe a couple days on maybe doing a follow-up task, but they might not be things where it’s like, “Okay, now I’m going to get into the nitty-gritty of the consolidation phase.” It’s like, “Let’s get the big ideas out.” And then as I follow up to that, I’ll have these consolidation days, where we then talk about some of the ideas.

Michael Rubin: I don’t have a great idea of how I’m going to do that yet, but I was thinking a lot of it is still going to be that curiosity, deeper thinking task base. Well let’s look at this. I was thinking a lot of … Are you familiar with Sara VanDerWerf?

Kyle Pearce: Yes. Yeah. We just brought here up on the last episode we recorded.

Michael Rubin: Oh perfect. Perfect.

Kyle Pearce: Jon referenced one of her sessions, so-

Michael Rubin: Yeah. I saw one of her sessions about the same time that I became familiar with you guys. She’s got this great line. Let’s see if I can not butcher it. It’s like, “Students will see it before I show them and they’ll say it before I tell them.

Kyle Pearce: That’s the line, huh. That’s the line Jon used.

Jon Orr: You actually said it better than I did, than I tried to reference it on the last episode.

Michael Rubin: And I actually have that at the top of, I’ve got my own little HyperDoc. I have that at the top of my HyperDoc, as the idea of being if I’m going to take this task and I’m going to try to consolidate it, I don’t imagine myself sitting there and saying, “Hey, why don’t we try this?” I imagine myself posting a solution that somebody else did and say, “Hey, this person or this or student did this. What did they do? Why did it work?”

Michael Rubin: Compare and contrast it to what you did, and have them think about those things, so that even in that consolidation, I’m not just talking at them. And so I’ve got that idea of this purposeful practice, and so when I look and say, “Oh, this is like a great purposeful question.” It’s like I’m not going to spend a bunch of time revisiting this concept. I’m not going to spend a bunch time hashing this out. This is something that we can extend with our thinking, maybe bring it up once or twice, but it’s within the main idea is an extension but it’s not something that I need to vote three days in a row to.

Michael Rubin: And then the other one was a concept throughout, where I realized that some of these ideas would just naturally come out in the task in the purposeful practice of everything else. So I didn’t really need to focus on those as things to find activities or tasks for either.

Jon Orr: What’s an example of one like that?

Michael Rubin: I can create linear expressions and equations in one variable that describe quantities in their relationships. So in my mind, students are going to be doing that when they’re doing a task. Students are going to be doing that when they’re working with sequences. Students are going to be doing that when they’re working with linear relations and systems of linear relations. That’s just something that’s going to carry throughout, so as I come up with the tasks and I’m trying to be intentional.

Michael Rubin: And it’s like okay, I might go down to something that I actually care about like a key idea. I can solve linear equations in one variable using the distributive property. I’m going to sit there and say, “Okay, well I don’t just want to give them an expression with the distributive property. I want them to create a linear expression using the distributive property in one variable, and then use what they know about that to solve it.”

Michael Rubin: So I’ll be referring back and that guides me to say, “Okay, how am I going to teach this task? What am I going to come up with? How am I going to present this, and it’s something that I will be referring back to.” My general plan is probably to print the list out and as I hit them, do a little check mark, so I can keep track of what I’ve covered when, to make sure I’m gathering all the ideas and they’re all coming out. That’s my general plan with this.

Kyle Pearce: Yeah. Very cool. I’m seeing too, as I’m looking through your list of dos, a lot of these are really awesome and I feel like over time, they may as you go through this, it might lead to more understands because I’m looking at for example, the do. For people watching in the academy, the 5.D.5.D which I believe is your fifth objective, it’s a do and it’s the fifth do and it’s A, B, C, D.

Kyle Pearce: So you’ve got this structured and organized really well, probably way easier for you to navigate than maybe someone else who comes in for the first time. I look at that one and I see and it says, “I can use the structure of an equation to help me solve the equation. For example, 2X plus 1 all over 3 is equal to 5/3rd means 2X plus 1 equals 5.

Kyle Pearce: Even just this idea of I’m picturing a student who understands what that’s all about, would be a student who could actually model both sides of the equipment into three groups or it’s 1/3rd the size on either side. And this other equation is three times bigger, so that tripling and thirding idea exists there that we can pull out number talks.

Kyle Pearce: So there’s so many things here that I think because you’ve organized this so well, that over time you’re going to probably see some of these ideas maybe flip-flop or maybe mirror in some ways in different areas of the no, understand, do part. Because I always found for me, I was big on the know part for a really long time, and then after some time I started shifting some of those into the dos, whether I realized I was doing this or not, and now I’m really heavily into the understands part. And the know and dos come out through the understanding in a lot of ways.

Kyle Pearce: So just something that I’m noticing, I have my own little epiphany here as I’m going through this document. And I feel like you’re the type of person Michael, that as you go through and you have these little epiphanies as you go along, I can picture you having no problem with pivoting from that plan and maybe revising or creating this new version of the document and doing the revisions over there and changing it for next time.

Kyle Pearce: So I want people who are watching and listening, to be getting the sense that you don’t have to have a finalized document or plan that you’re going to do for the rest of time, because that will really, really limit you in how effective you can be. I’ve worked with some educators, amazing educators who I would call them leaders, math-content leaders in my own district.

Kyle Pearce: And they’ve done something similar, not from a spiraled approach but from a scope and sequence approach, and they’ve put so much effort into it, but now I feel like they’ve gotten to a place where they’re stuck because they don’t want to change anything. They put so much sweat, blood, tears, not so much blood but the tears and the sweat, into this document that now they feel like if they make any changes they’ll ruin it.

Kyle Pearce: And I think that’s something that we want to make sure people at home know is you don’t have to start this process and feel like it’s got to be complete or in its final draft the first time you use it. It’s a process I think, of learning. Do you feel like this has helped you better learn your course content from just engaging in this process, Michael?

Michael Rubin: Yeah. Absolutely. And I’d also like to echo what you were saying with that, and you talked about this on the last episode I was with you too well, about flexibility. And as I was creating these, I had this idea that as I go through the year, I bet more things are going to come out and I’ll just add them and that will be totally fine because I’m one person and I don’t have a buddy I’m working with on this. So it’s like I know there are things I’m going to miss, especially as I’m going through and doing these from my other courses as well, where I’m not as familiar with the content. I realize okay, I’m going to be able to add things to it.

Michael Rubin: You also just answered one of the questions I think, that I was going to ask you later, which is do you spiral by those essential understanding or do you spiral by the skills? And so it sounds like you were saying that you try to spiral by those key understandings and I’ve gone the other direction. So I thought it would be a lot easier in terms of organizing by task or organizing on how I’m going to teach to do it by skills, and just keep in my that as I’m doing that, I need to focus on those essential understanding, so I thought that was interesting.

Michael Rubin: From a beginner’s perspective it seems like going through and saying, “Okay well, we’re going to solve simple linear equations first and then we might bring up the distributive property and then multiple terms on a side, combining terms and then variables on both sides.” That would be like a natural progression where these equations get a little bit more complex. And that as I’m spiraling I can start with just the simple ones and the next spiral through I’ll introduce something with maybe multiple terms, and then after that variables on both sides and then after that the distributive property.

Michael Rubin: And seemed for a novice to be easier than say, “Well, I’m going to focus on expressing relationships and then I’m going to focus on equations as a process of reasoning and explaining that reasoning. I don’t see how to do that yet.”

Jon Orr: I think what Kyle is saying is that we’ve said that already too, it’s like there’s no really one way to do this, and like you said that you are ready to modify this whenever you need to. And I think I have revamped my plan for my grade nine class every semester. I have an outline of the way I think it might run but then it doesn’t flow that way because of the pivoting in the moment. So we’ve talked about different ways to spiral in the last one. Do you spiral by Kyle suggesting or you can spiral by a big idea and then think about always those big ideas.

Jon Orr: Some people spiral by the conceptual and then move to more specifics afterwards. Think about one teacher was spiraling everything by doing graphs and pictures. Everything was represented by graphs, pictures, concrete materials and then the next time was like, “I’m going to slowly integrate abstract representations eventually, and by the end we’re doing those same ideas over and over again but we’re getting more specific.” That would be big idea down to more specific skill sets, and starting with skill sets and then going big ideas, also can work and there’s nothing wrong with that idea.

Jon Orr: I think the biggest piece of why spiraling is so great is doing what you’re doing right now. As Kyle said, you are becoming a master of the curriculum right now. You are understanding so much more about the connections between math concepts as a teacher, and all that is doing, you making this plan is preparing you to be flexible on the fly. You’re going to be able to know the connection.

Jon Orr: You’ve done all the hard work to see these connections in the curriculum that you have to teach, rather the standards that you have to teach, so that when you’re in the classroom, and see a student doing a solution here, you know where that fits in the plan. And you’re going to be ready to modify whatever you need to do on that. I think that is the biggest, in my opinion, takeaway of trying to spiral your course is you’re becoming a master of the math and that’s going to be able to help you in the classroom.

Kyle Pearce: Just to echo what Jon’s saying as well, I think you just nailed it. I picture eighth overall objective here, this algebra objective and it involves systems of linear equations, inequalities and two variables. My apologies if you’re an early elementary teacher and you’re thinking, what the heck are we talking about? But the reason I pick this one was because it is like you’ve got it chunked down into the least number of parts here.

Kyle Pearce: I think Jon just mentioned this idea that how you choose to frame or organize things, the one thing that we just don’t want to forget is that the end goal is that overall objective at the top there, right? So that’s a big thing, and then when I go one step removed from that I go, when I say solve systems of linear equations, what are kids thinking about? What are they conceptualizing and what is their conception of what that is all about?

Kyle Pearce: In their mind, could they tell me that, “Oh okay, so you’re talking about a scenario where there’s like two taxis. One taxi starts with this flat rate and then it increases by this amount. And this other taxi doesn’t have a flat rate and it increases by this amount. And we want to know essentially, when one’s the better deal over the other. To me that’s like a conception about solving linear systems of equations where that student’s going, “I get what we’re trying to accomplish here.”

Kyle Pearce: And then I think when you go down a step deeper than that, so we take one more shovel out. Then we start to go, “Okay, so how can I actually do those things?” And I guess my only caution for folks is to avoid what I did for a very long time, which was trying to check off those individual skills, and at the end I tried to bow them together. Tie them together under this big clump to go like, “There is your overall objective.”

Kyle Pearce: Whereas I think by doing so, it was harder for kids to take these separate pieces and glue them together. Rather than starting with big idea piece, which we just described what that big idea is. So we’re like, “That’s it. Hey guys, gals. This is all we got. This is the big thing that we want to do.”

Kyle Pearce: Here’s the hard part. The hard part is there’s so many different ways to attack it. It’s like look at our conversation about spiraling. Spiraling is this. We dig deeper and deeper and deeper, we don’t want to forget the overall goal of why we wanted to spiral in the first place. It’s so easy for me to dig deep enough, and then all of a sudden I find myself trying to mix as many concepts together as possible because that’s what the word spiraling reminds me of is just this blender.

Kyle Pearce: Hit the blender and now it’s spiraled but what a second, what am I doing this for again? Oh yeah. I’m doing it because I want to make sure that the concepts are accessible by all students in the class and for as long as possible in my class so we don’t dig too deep, too fast. I want to make sure kids retain it, meaning that I’m now giving them an opportunity to come to ideas.

Kyle Pearce: And I also look at skill piece as whenever I come back to an idea, regardless of the choice of how to spiral, I come back to an idea, but then I start a step below where students are ready. Meaning, I’m going to start a little bit behind where we left off last time, and it’s not going to consume a lot of time. It’s going to actually allow it make sure everybody’s on that bus with me, or everyone’s on that elevator with me and we’re going to go up a level together.

Kyle Pearce: Instead of saying, “Well, this is where we got off the elevator last time, let’s everyone get on.” When in reality we’re like, “Wait a second, a bunch of kids are not on that floor. They either didn’t retain it, they didn’t get it the first time. Whatever it was, now I’m giving them that second opportunity to practice some of those skills, and then those skills come along the way.

Kyle Pearce: At the end of the day if a student doesn’t know how to solve by graphing, substitution and elimination, those are three big ideas, its like can they do one of those? That’s way better to me, than doing all three of those and not realizing why they did any of it, which was my whole beginning of my teaching career. They could do those three things but they didn’t realize that you could have used any of those three in all those scenarios, or that one of those was way easier to do in this scenario than in that scenario. So I know that was a mouthful but just some things to think about anyway.

Jon Orr: I was going to bring that example up because that skill about solving systems of equations, I remember teaching it exactly that. It would be like it would go over your head. We’re going to solve where two lines cross by graphing, and then we’re going to solve by substitution, and sometimes kids see the connection between those two and don’t, and now I’m going to solve by elimination. Then you’re going to go, “Hey, by the way, that was three different ways to solve any problem where you had two equations and two unknowns. And now on our last before the test, we’re going to throw them all in. Now we’re going to say, “Which way would you do it now?”

Jon Orr: It makes a huge amount of sense like you said Kyle, to start with the whole idea of I want to figure this out the way I’ve modified it this last few years I think was based on some of the work from Alex Overwick and the first lesson is like you said also really low so that everyone can do it. It’s like, “Hey, there’s four jujubes and three Smarties at this store and they cost this much cents. And the same store there’s three more jujubes and five Smarties that cost this much. Here’s a whole bunch of pennies. Here’s the pennies themselves. Here’s something that could represent jujubes. Here’s something that could represent Smarties. Take the pennies that those combination is supposed to add to for the one situation. Here’s the pennies for the other situation. Put the pennies on the counters. Put the pennies on those shapes and make it work.”

Jon Orr: What are kids really doing there? Very concrete. Very hands-on and then slowly start to build up into those other things. Start with that big idea about we’re trying to figure out what these values are in these situations. Then by the end you don’t have go like, “Oh okay, now we’re going to do the problem-solving day where we have to decide.” You’ve been deciding the whole way through to begin with.

Kyle Pearce: Yeah. I’m just playing in the background here. Jon I don’t know if you see it but just sharing my screen showing Accounting Candy Sequel. And that first video doesn’t show a whole lot, but it just gives you a quick glimpse of three groups or three colors of candies, and then we zoom in on it so you can’t see it. And then basically the big question that kids are really having to figure out is listen, if I know there’s 57 chocolates in total, they’re just colored different colored candies. I know there’s 44 blue and pink, there’s 35 blue and purple and then I’ve just got a summary of all three of those pieces of information by how many of each color are there?

Kyle Pearce: And I usually use this task way before we even introduce this idea of solving systems of linear equations at all. And what happens is rather than me teaching it, what ends up happening is kids do all kinds of really cool solutions strategies. They just use basic arithmetic to figure this out. Students take the total candies. They subtract 44 blue and pink, and they know that’s how many purple there must be.

Kyle Pearce: So right away, they’re using this idea of elimination without even realizing it. They’re even using some substitution in some cases. And meanwhile for me, I would have done it this big, long way. When I show on the task side I’ll include it in the show notes, but you see this big, long solution where I’m taking stuff and substituting this into that and this into this. And students have done this in 13 seconds flat because they have that understanding of what it is that they’re trying to achieve.

Kyle Pearce: So that just popped into my mind as Jon was sharing there, but I’m wondering we could pause for a moment and get your thoughts there Michael, on this idea of starting with the idea of spiraling and how you want to be thinking about organizing, and then also just in terms of the intentionality of the content that you’re going to be delivering.

Michael Rubin: Yeah. Perfect and that might a great segue too. So first of all I’ve just took a page of notes on what you just said and wrote “takeaway” because you always ask what your key takeaway and that’s a stressful moment. So I’m writing this down. I really appreciate that idea of always going back. That was an idea I had, not something that was forefront in my mind that whatever skill I’m working on, make sure it’s going back to this main concept and that students understand how it’s all connected. That is my intention, but I also see how it could be really easy to say, “Oh well, now we’re solving these linear equations by graphing or linear systems by graphing or whatever, and now we’re going to solve them with substitution and have them unrelated.”

Michael Rubin: I had this idea as you were speaking, one really good way of doing that is to maybe use the same context. I can see you’ve got something that we’re graphing as a nice way of solving it, and then just take the same context, modify the numbers where they just don’t work out where the students try to graph it.

Michael Rubin: I don’t even know. I might give students this problem and none of them might try. I might give them an introductory test and none of them might graph, so that might a place where I pivot and it’s like, “Okay, so now we’re talking about these types of problems.”

Michael Rubin: My plan too, now that I’m thinking of it, I really like Jon’s idea of these mastery days. And part of my plan with the mastery days is that’s going to be the day when we go through and I formally introduce the content objectives where it’s like we were working on solving systems of linear equations and we can talk about what does that mean, and what were some of the different types of problems we solved and how were they connected? How were they similar? How were they different? It seems like those might be really great days as well, to make some of those connections. So I have those two ideas.

Kyle Pearce: I just wanted to make sure before we move on, just the idea of really taking advantage of context and not trying to overdo new context all the time. I did that forever and even before pre-transformation for me, when I would go and I would try to make question for kids to do in my class, I would try to make 13 different contexts every day. It was like this problem and this problem that’s completely unrelated and this problem.

Kyle Pearce: It’s like I spent so much of my own planning time, and then also teaching time, helping kids to wrap their heads around a new context, rather than taking that one context, spending that little extra time up front to get them to really understand it, and to feel like they owned the context, and then building ideas off of that. So you can juice them, just totally wring them of all that they’re worth, and not worry so much about …

Kyle Pearce: Curiosity to me is really important at the beginning of an idea, beginning of a concept especially when you’re culture building your classroom. But as you go and as trust builds and kids are comfortable in your non-threatening classroom environment, I feel like sometimes you don’t have to rely on that so much because now it’s just like they know what they’re here for. It’s like there’s no fear. We’re here to dive into some really, really important concepts and it’s not just about answer getting, it’s about math and discovering how math works and fueling that sense-making. Jon, what were you going to say? Hopefully I didn’t just steal your entire mouthful there.

Jon Orr: Well, I wanted to talk about the mastery days and just when you said about the making consolidations on those days. That’s a great idea but I just want to remind you, don’t forget to, it’s so important that on the day you tasks or the day you do a learning objective or a learning goal, to make sure by the end of that, we’re going to talk exactly about what we did then too.

Jon Orr: So it’s like when you said it, I got the image that you’re waiting until a mastery day to talk about, “Hey, remember back on Monday when we did this thing? Oh, what we were doing was this thing over here.” I think I would want you to do that both days. It’s like do that on the day you’re doing it, so that’s like we’re consolidating on what we’re really doing here. What is the point of today? Dan Myer, I remember seeing him and he said that one of the things you could do at the end of a lesson was he had kids title the lesson.

Jon Orr: People always want to put up front what the learning goal should be at the very beginning of a lesson. And we talked many times I think, on the podcast about withholding that a little bit until later, and at the end is a great time to do that. Like, “What is the learning goal here today? If you had to title this lesson, what would you title it?” And then you can help shape what that title might be with attaching the learning goal to it.

Jon Orr: It’s like asking someone their big takeaway like, “What did you get out of this lesson today?” And making sure everybody walks away with, “Oh that was related to yesterday’s work.” Or “That was related to the task we did a month ago.” So but it’s also a great idea to do it on those mastery days where you spend consolidating some stuff.

Kyle Pearce: That’s a great point Jon. I didn’t even think about that piece but now that you’re mentioning it, I know for a fact that there was many years I was teaching where I thought that kids were getting it through the inquiry process, but oftentimes that’s not the case. There’s some kids that completely miss the point. It’s like when you’re watching a movie and you’re totally in this movie and you totally get what’s going on, and then the next person nudges you and says, “Wait, who is that person on the video?” And you’re like, “It’s the bad guy. How obvious to me was that but to you it wasn’t?”

Kyle Pearce: I think in math class it’s even more so. It’s so much easier for me to drift. I’m sitting in class and I just drift off for just even a moment, and I miss that key piece. So I think having a starting and an ending each day, each lesson with some consolidation, to make sure that there was very explicit as to like, “Hey friends, this is why we did what we did today.”

Kyle Pearce: It’s great when a student comes up with it and they can share it, but then you’re still reiterating it. So if you can use student work and use those five practices to select and sequence the work and share. Letting three students share, not everybody share but then you taking that and going, “Okay, based on what this student said, I’m going to rephrase what they said to make sure that everybody’s clear because maybe they weren’t clear on how they said.”

Kyle Pearce: That’s the other thing. I hear a student say something and I connect the dots because I already know what they’re trying to say. What about the kids who don’t know what they’re trying to say? So I have to sometimes restate what the kids are saying, and then I go to this group and I let them share and I say, “Hey, can you just share about this part over here because we already know about this part. I really liked your strategy over here.” Maybe it’s that graphing idea that came out. They share that and then you can share something you liked about it but then maybe extended a little bit further.

Kyle Pearce: And then you go to that third group and then finally you have to think about okay, now given what they’re sharing, what can I do to help and actually deliver some value here? I definitely don’t want people to hear Jon and I speaking, thinking that Kyle and Jon only sit on the side or meddle in the middle like Jennifer Waddle said on the recent episode there Jon. I’d love that but I think the consolidation though, is one of those times where we can’t be fearful to say what needs to be said, clarify what needs to be clarified.

Kyle Pearce: Otherwise what I think can happen and what happened in my classroom for many years was some kids got it but a lot of kids didn’t. And then the result of that is that people call the type of learning in that environment, discovery-based math curriculum. And people will say, “Discovery math is a poor way to go.” And I’m going to agree. Discovery math with no intentionality and no clear, explicit explaining of the concepts is aimless for a lot of students.

Kyle Pearce: And again like I said, I did it for years. I know there were kids who left and I wasn’t clear, concise enough, explicit enough and partially was because of my fear of no, I’m not allowed to do this in this style of more progressive teaching. That was one fear, which I think is a misconception, but then the other part was I don’t think I really understood the mathematics well enough, and that’s what I was referencing earlier about a lot of those dos.

Kyle Pearce: As you start to continually dive deeper into the concepts, a lot of those will start to become concepts and things you want kids to be able to understand and really own. And as I started to own more of the math learning, I was able to help students explicitly see those concepts and pull at them. So any thoughts there? I saw you writing a few things down, so we’ll let you share your thinking and Jon and I will settle down for a moment.

Michael Rubin: Yeah. I think I just wrote about four pages. So first of all I’m a math nerd and more specifically a teaching nerd, so when I say this I hope people don’t judge me, but literally my adrenaline’s pumping right now with these ideas. I’m so excited because that’s something I’ve really struggled with and I tend to be an all or nothing person. So this idea that yeah, no it’s okay to consolidate and consolidate and specifically tie to the big ideas.

Michael Rubin: It reminds me of a colleague who just this last year I was working with their, she was doing an inquiry task with drawing right triangles with the same angle to explore some trig ratios. And so she asked her students at the end of the class like, “What was the purpose of this activity?” And a kid raised their hand and they were very proud of themselves and they said, “We drew triangles and measured the side lengths.” It’s like they completely didn’t understand what they were doing or why they were doing it.

Michael Rubin: So I actually just as you were doing that, I’ve got this overall template for each day and I added in what the concept introduction day is the mastery day, it’s consolidate days. I wrote in, “Consolidate, tie to big idea. Why are we doing this? What are the concepts?” That this isn’t something where you have the … I was thinking about having students write a reflection. If I click there Kyle, on my day-to-day in that green-

Kyle Pearce: This one right here?

Michael Rubin: … you can see my … Yeah. So here’s a link to my year long day-by-day plan. Yeah. There you go.

Kyle Pearce: Beautiful.

Michael Rubin: Initially when I was going to spiral by the way, I was just going to do it like this and I realized I had no flexibility in it. So I pivoted once again but I’ve got these, this is where I’m going to link my lessons, put all the links and stuff so I have it all organized and I’ve got these big ideas. I’m going to have an engaging start every day. I’m going to teach through some sort of task, maybe some purposeful practice or formative assessment, assessment for learning, reflections, exit tickets. On some days a gut community building and mindfulness tapped in there as well, and so I just added that consolidate tie to that big idea.

Michael Rubin: That’s really exciting for me because I think that’s something that I was and I probably still will struggle with a little bit, especially with having a lot of English learners. Again, I got really excited about that idea of juicing that context. One that I was thinking of was the stacking paper because you can take that and disproportional reasoning all the way through systems of equations. You just have a different height table and different stack sizes of paper and have them figure out at what number of stacks will they be the same height? And you can use that context and I think tying that context will also help them realize, oh these are the big ideas. This is how they connect. This is the purpose of what we’re doing. How is this one different from the last one?

Michael Rubin: I love that idea of starting a little bit behind. It also answers another one of my questions like, “What do you do for the kids who missed a day or missed that task day or came in part way through the year that’s starting that little bit behind get to them caught up as well?” And then it makes those connections explicit like what was the strategy here? What was the strategy there and what were the main ideas? That was incredibly helpful.

Jon Orr: It brings up a lot of ideas for me, specifically lately about what notes look like or the end. If you’re going to consolidate at the end, what does that format look like as a takeaway for students? You had mentioned a reflection but what I’ve been thinking about lately is especially for my applied level students, what does that note look like are probably terrible at taking notes themselves, not dedicated enough to write notes down. They’re disorganized.

Jon Orr: I was spending a lot of time lately on what should that takeaway look like, so that students have something to look back on if they need to, because for a long time I was like, “If they do a lot of work, I still believe this, if they do a lot of math thinking and math doing in my classroom for my hour and 15 minutes with them, do they need to take any notes away to reflect on and to look back on?”

Jon Orr: In the higher levels I get I’m thinking, yes, I think they do need something to not necessarily look back on but the act of writing it down I think is important. And they can reflect on it later but it’s like what is that takeaway at the end, so when they write it down. We have big idea written down. It’s connected to the big idea we looked at before. There might be an example written in there. I’m really concerned or interested in what people’s notes look like now. I used to tell my kids, “Oh, you should take pictures of the work that they did.” And that acts as their note but I still think that there’s so much value in consolidating the learning they did, that they can take away.

Jon Orr: I’m really interested in the interactive notebooks that I’ve seen floating around the Internet and resources. Some of those are the foldables and the flip books, some of those can act as a great resource. I’m really interested in that. I’m still exploring what is the best way that a consolidation piece at the end of a lesson looks like, so that there’s something to take away.

Kyle Pearce: What do you think Jon?

Michael Rubin: Yeah. I don’t know if this is something that you’ll care about at all, you might just want to cut this from the thing as it’s not really related to spiraling but it’s something I’ve worried about. And as I’ve been trying to incorporate all these new things, that reflection piece was the big idea, so I just in the chat notes I linked two documents.

Michael Rubin: We have, our district hired a company, I think it’s called E.L. Achieve and they do this thing called constructing meaning. And so I went to this workshop, which is all about providing academic language supports, so they talk about the bricks and the mortar, meaning that you’ve got your brick language which is all the vocabulary that you do in class like the math vocabulary. But then you’ve got all this connecting language that kids don’t necessarily know how to use, especially English-language learners. So do you see the links in the chat notes?

Kyle Pearce: Mm-hmm (affirmative).

Jon Orr: Yeah.

Michael Rubin: So if you click those and open those, one of them I use, I don’t know if I’m allowed to talk about their stuff but I adapted it from what they did. So if you click on the other one first, you can see how I’m planning on doing this.

Kyle Pearce: We’ll include a link. I just pulled up elachieve.org, so I mean you’re properly citing and we’ll throw it in the show notes, and then the academy as a link as well as a resource people can look at, so I’m sure they’ll appreciate that.

Michael Rubin: Okay. So what they do is in that paragraph writing, they don’t know how to do that. And so I’ve come up with this and it gets to the idea, and the question I took specifically out of Jo Boaler’s Mathematical Mindsets book where she had her reflection question, which I initially was going to just throw at the kids. And as Jon was saying, I realized kids aren’t going to necessarily know what to do there. They’re not going to know what to write, and so you come up with this idea of the paragraph and the kids can literally, they don’t need to use it but if they don’t know how to start or what to say, they can use this framework where it’s like the main idea or concept of in the check box is lesson, problem, activity, discussion or other thing.

Michael Rubin: And then you’ve got the verb, examined, addressed, introduced, supported, explored and then the next box. The idea of, the relationship between, the importance of, the reasons for, the difference between. And then they can fill that in and explain with the content language and it goes all the way through. They introduce. They explain and describe. Support with key ideas. So blank is a key detail or is an example or is an essential information that illustrates or that explains that suggests blank because blank.

Michael Rubin: I’ve actually gone through and I just took a topic and wrote a couple of examples of what those paragraphs might look like. This is something I planned on explicitly teaching with my students. So the first time we do this we’re probably going to do it together, and it might even be the first 10 times we do it together. And I was thinking this would be a way that students could even if they feel like they have an exemplary one, they can share it with me and I can share it to the class through Google Classroom. So it’s a way for kids to consolidate that information.

Michael Rubin: As I was doing this though, I realized Jon was saying as well, is that even if you have this great reflection, it might not include everything. So in the reflection and notes document, which was the one you had linked to before Kyle, I’ve got the topic and that’s the title of the lesson. What’s the topic? I don’t fill that out for them. It’s like what would you title this lesson? You’ve got the space for the question, their response and then any other notes that they feel would be relevant. So they could put in an illustration, an example problem. Maybe if they didn’t write it out, an efficient method. Maybe a side comment that a kid made. An alternate strategy that they’ve got a lot of flexibility in doing that.

Michael Rubin: And this might be a place whereas we work on this, either the day of the task if the task is short enough, I only have 57 minutes, or the next day during a consolidation phase we trade these things. We have kids write feedback on them. We discuss something that maybe was helpful that they found in the students’ notes and we used that to get to the main, big ideas as well. I mean I don’t know if this is going to work, I’ve never done it but I felt like I needed a plan.

Kyle Pearce: No, that’s good. All right so Michael, we’ve been doing a lot of chatting here. This has been super, super helpful. I’m in this tab here with it looks like your objectives that are summarized a little bit here. You’ve got a list of tasks and objectives. What did you do next with those, in order to start framing out what this is going to look like, as the new school year quickly approaches for you in California?

Michael Rubin: Yeah. So where we left off at the last discussion that we had, I’d started looking for tasks and I figured I needed a way to organize them. And so as I was going through tasks just searching websites, I created a list of all my main objectives and started just linking them in, in no particular order, and I realized to spiral I was going to need a little bit more organization than that. So I took the progression idea, and on that second tab I broke down every topic like algebra, geometry and statistics and I broke them down into the main progressions and the main ideas.

Michael Rubin: And so once I did that, I was able to sit down, write out any prerequisites I thought kids might need in order to deal with those. And then go back to that first tab where I listed tasks and figure out, well where do they fit in here? It’s not that I’m going to necessarily use them all, especially as we’ve talked now about juicing those context, but I have a menu of tasks to use and I’m sure I’ll be adding to this list as we go along.

Michael Rubin: And then came the task of actually spiraling it and this is where I had some concerns. I’ve learned a lot in our discussion today, but on that third tab one of my biggest concerns has been our benchmark assessments because the way that we teach it even though it’s integrated, we really spend the first semester and we do algebra and we do the second semester geometry and statistics. And so I know that all my colleagues are going to be getting to all these, going in depth into all the algebra concepts right away. And so I had to figure out how was I going to spiral these out, while at the same time preparing my students for these benchmark assessments?

Michael Rubin: So I work on a compromise, so instead of saying, “Okay, we’ll I’m just going to do every topic.” All the basic concepts of algebra, geometry and stats and then just move down equally, I decided to try them or whatever. So I highlighted the ones that I’m focusing on and that’s going to roughly correlate to what the rest of the school’s doing at that time.

Michael Rubin: So if you look at, you’re looking at the screen here. I’ll give a quick description of the spreadsheet for those that are listening. I’ve got five columns and the first one’s just a cycle number, so that’s going to refer to maybe a two, two and a half week cycle. I’ll be flexible with that. And then I’ve got an algebra concept, a geometry concept, a statistics concept and any other topics that I think should be revisited or reviewed either through purposeful practice or maybe some homework problems or warmups. And I figured these are going to be the main ideas I’m going to take out it.

Michael Rubin: And the process I went through, is I went through the progression and I figured out how do these progressions build? And I started with the simpler ones and just started building them in until they became more and more complex. Then I went through, I did the same thing with geometry, realizing that we’re going to want to hit some of those bigger ideas more in that second semester when we’re going to have those benchmark assessments. So I saved some of those big ideas of congruence and triangle, angle relationships and proof.

Michael Rubin: Until then I like the idea of introducing those concepts earlier, but because I’m going to have to spend a little bit more time on algebra, so that my students are ready for the final and the benchmark assessments, I had to make some choices about what I wanted to do. So that’s how I organized my spirals.

Kyle Pearce: So clearly the amount of planning you have put into this is so key. So if someone is listening at home and they’re like, “Well, I want to try doing this.” And again when we say, “Spiraling” it’s not like a this or that, it’s like it might be dipping your toes in and you might just make certain modifications. But I think all of this preplanning you’ve done is huge and I think it probably makes this process of actually trying to organize things, make so much more sense.

Kyle Pearce: I really want to also reference how you’ve taken things and you’ve really tried to show them on a bit of a progression, which is great, or a trajectory as some people might call it. And you had also mentioned earlier that Jon and I are big on really thinking about the mathematics of where does it come from and where is it going, and trying to figure out how am I going to organize this along a trajectory? It doesn’t necessarily all have to happen one after the other, but we just want to make sure that it’s in at least a reasonable order as things are happening. And then how do those trajectories or those progression interconnect if any?

Kyle Pearce: So super cool. I love that and then I love this spiraling plan as well, where you’ve got things, you’re focusing on different things as you go. And I’ve got to assume and again it’s an assumption that you’ll find out after this year as you do this, but by introducing things and not doing all the algebra at the beginning and then all the geometry. I think that’s going to have some positive results on those benchmarks later because it’s not like it’s all gone and then you don’t see it for X number of weeks while you’re doing geometry.

Kyle Pearce: So I think that constant mix is really going to serve students well, despite the fact that you might not get as deep into certain things as maybe you might have in the past. So it will be really cool to see how that goes. Jon, any reflections on that?

Jon Orr: Yeah. No, I’m impressed with the level of organization, for sure. I’m thinking back and comparing the way we did it when we outlined it in the last episode. When we just did it, we didn’t have anyone to talk to about it, we just shot from the hip. So it’s very, very nice to see the level of understanding of the concepts going into your planning. I think as I said before, it’s so key to think about that and I think if anyone gets anything out of this conversation, I think it is about understanding the connections amongst the concepts, and it’s necessary to think about that before you start say, making a planning document yourself.

Jon Orr: So great job here. We are impressed and as Kyle said, we’d love to just talk to you after you’ve taught this and see what things you’d change, what you learned along the way. So it will be great to have you back again on another episode next year, once you’ve put this into place. I think that would be great for people to listen to.

Michael Rubin: Yeah. I’d also like to reiterate on that planning. For somebody that just did it, I could not have done this without the planning. I also felt that the planning was so important for me as I was making decisions. In the past we’ve had these geometry units and it’s like, “Oh, well we’re doing constructions for two weeks.” And there was no reason for the constructions. It’s like, “Okay, well it’s just part of the unit.” And now it’s like, “Oh well, really we’re going to be doing these constructions.”

Michael Rubin: It’s great for problem solving, especially if you problematize them. It’s like, “Can you construct this and have the kids work collaboratively to do so?” That’s great but then really getting that idea that you can use this concept of constructions to establish the triangle congruence postulates and that we’re setting them up for that.

Michael Rubin: So the reason that we’re spending all this time on the constructions is so that they’re ready to engage in maybe some more meaty concepts of proof and justification. And these are understanding that I’d never had before, so being able to go through and ask myself, “What are the key concepts? What are the big questions? Why am I doing this?” This has been super helpful in this plan.

Kyle Pearce: Michael again, I think you’ve done so much work in knowing and being, I don’t want to say a part of the whole journey because we definitely weren’t, but you’ve done a lot of this work on your own. But the one thing that I want to make sure that people are aware is that you didn’t try to do it completely independently.

Kyle Pearce: So right from the beginning when you were in the workshop last fall, there was lots of talk about it in the discussion area. Currently in the academy, you’re still right in there and talking about it and really doing a lot of co-learning with people out there. And I know that you’re going to be a huge asset to the rest of that Math Moment Maker Community.

Kyle Pearce: So a huge, huge thank you to you for being vulnerable enough to come on here, not just once but twice to share on the podcast. And as we had mentioned beforehand, we would love to be able to take the video we shot while we were having this discussion, so that those who are in the academy can really dive in a little bit deeper. And then hopefully others in the academy who are doing something like what you’re doing, and maybe they’re doing it in a different way, we’ll try to get them on for some episodes as well. Try to get different perspectives in here. We can all learn from one another.

Kyle Pearce: I know for me a big takeaway was starting with those big ideas like you had done in your planning process. You had spent, I’m going to say, 80% of the work was in the preplanning and the small amount of work is actually now where you place everything and the little adjustments and even the tasks. I know that’s a big too, but as more and more people are doing the work like you’re doing here and sharing that out, people will have at least a great place to know that okay, these tasks are great for these ideas. And here’s some really cool big ideas that I can start with, and then I’m just going to modify to suit my own context, my own class, my own course.

Kyle Pearce: So thank you to that or for that. Jon, any last minute comments here or are we ready to wrap this thing up?

Jon Orr: Man, I think we are ready. Before we go, we would love for you, Michael, I know that you were ready for this, do you want to chat about what you say your big takeaway was for today?

Michael Rubin: Yeah. I mean I feel like there were so many, but I think the one that really resonated with me, was especially in this idea of spiraling where you’re not just dealing with one big idea at a time. It’s how important it is to make those connections to the big main big ideas and the concept’s explicit so that students really understand that hey, we’re solving this problem today but this is similar to the concept that we studied two weeks ago. We’re just using a different strategy or a different method and these are the reasons we’re using that different strategy or method but we’re really solving the same types of problems. We’re thinking about the same ideas.

Michael Rubin: That was really helpful, that idea that we should consolidate every day, maybe not at the beginning of the lesson because you don’t want to ruin the surprise of the task but definitely by the end that you make those connections, so kids walk away understanding that. That was really helpful.

Kyle Pearce: Amazing.

Jon Orr: Great. That’s good.

Kyle Pearce: That’s super helpful. Well thanks so much again, Michael. We really appreciate your time and as mentioned we’re learning from one another. I know Jon and I have picked up a huge amount of ideas. I love the way you’ve structured your planning, so we’re definitely going to be keeping a copy of that for our own explorations as we move forward. And hopefully we will continue to see you around the community area of the Math Moment Maker Community inside the academy. And we will be up-to-date I’m sure, throughout the year as things go and as you pivot and continue planning and pivoting along your spiraling journey.

Michael Rubin: Absolutely. Sounds great.

Kyle Pearce: All right.

Michael Rubin: Take care.

Kyle Pearce: Have a great one. We’ll talk to you soon.

Michael Rubin: All right. You too.

Kyle Pearce: As always, both Jon and I learned so much from these Math Mentoring Moment episodes, but in order to ensure we hang onto this new learning, let’s be sure to reflect on the big ideas and big takeaways that we’ve learned. An excellent way to do this and help learning stick, is to reflect and create a plan for yourself to take action on something that resonated with you in this episode. A great way to hold yourself accountable is to write it down and even better, share it with someone, your partner, colleague or with the Math Moment Maker Community by commenting on the show notes page, tagging at #MakeMathMoments on social media or in our free, private Facebook group, Math Moment Makers K-12.

Jon Orr: Another great way you can take action is to take the action that Michael did and do a deep dive with us in our six-module, online workshop each spring and fall, where we coach you step by step on how to create and modify your lessons to engage students, build deeper understanding of math and promote resilience in problem-solving. With over 250 educators already through the program, we’re confident you too can create resilient, engaged problem-solvers just like Michael has learned how to do.

Kyle Pearce: And if taking a deep dive isn’t enough, the workshop is accredited for two to four PD credit hours through Brandman University. If you are interested in learning more about registering, be sure to check out MakeMathMoments.com/OnlineWorkshop. Remember registration closes soon, so head over to MakeMathMoments.com/OnlineWorkshop.

Jon Orr: Are you interested in joining us on the podcast for an upcoming Math Mentoring Moment episode where you too can share a big math class struggle? You can apply over at MakeMathMoments.com/Mentor. That’s MakeMathMoments.com/Mentor.

Kyle Pearce: In order to ensure you don’t miss out on any episodes, be sure to tap the subscribe button in your favorite podcasting platform. And if you like what you’re hearing, do us a huge solid and be sure to share the podcast with a colleague and help us reach a wider audience by leaving us a review on iTunes, Spotify, Google Play or whatever platform you are listening to this on. Every bit helps.

Jon Orr: Show notes and links to resources from this episode can be found at MakeMathMoments.com/Episode42. Again that’s MakeMathMoments.com/Episode42.

Kyle Pearce: Well until next time, I’m Kyle Pearce.

Jon Orr: And I’m Jon Orr.

Kyle Pearce: High fives for us.

Jon Orr: And high fives for you.



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Thanks For Listening

To help out the show:


  1. Tara

    What was the name of the book he referenced early on during the podcast? I couldn’t find it in the show notes

    • Melissa

      Routines For Reasoning



  1. Episode #52 The Purpose of Assessment: A Math Mentoring Moment - Make Math Moments - […] 40 and 42 with Michael Rubin on […]

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