Episode #40: How Do I Know When To Proceed Or Pivot My Plan? – [Math Mentoring Moment]

Sep 2, 2019 | Podcast | 0 comments


This week we speak with Michael Rubin, a Math Moment Maker who has been diving into a TON of Math Professional Learning over the past year as he gears up to make a return to the math classroom. Join in on the conversation as Michael shares his recent successes learning from the many math educators sharing online, our weekly tasks/tips/tools weekly email, as well as some epiphanies he’s had since taking our Online Workshop recently.

You’ll Learn

  • How Do I Teach Through Task?
  • What Learning & Preparation Do I Need To Do?
  • What Strategies Can Help All My Students?
  • 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?


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Michael Rubin: I had several main concerns that you talked a lot about. Is that while that one student wouldn’t leave for lunch, I had another student who was the student that typically came into class, paid attention, figured out the procedures, solved the problems. And this student had a meltdown. I gave that same task and they didn’t know what to do, and they just had this meltdown. And I didn’t have any guidance to know what do I do about that?

Kyle Pearce: You’re listening to a snippet of our conversation with Michael Rubin, a Math Moment Maker who has been diving into a ton of math professional learning over the past year as he gears up to make a return to the math classroom.

Jon Orr: Join in on the conversation as Michael shares his recent successes learning from the many math educators sharing online, our weekly tasks, tips, tools, weekly email as well as some epiphanies he’s had since taking our online workshop recently.

Kyle Pearce: Are you ready to dive into another amazing math Mentoring Moment episode?

Jon Orr: I know I sure am. And guess what? There was so much goodness happening in this conversation that we had to what split it into two parts. Hit it.

Kyle Pearce: Welcome to the Making Math Moments That matter podcast. I’m Kyle Pearce from tapintoteenminds.com.

Jon Orr: And I’m Jon Orr from mrorr-sgeek.com. We are two math teachers who, together-

Kyle Pearce: With you, the community of math educators worldwide who want to build and deliver lessons that spark engagement-

Jon Orr: … fuel learning-

Kyle Pearce: … and ignite teacher action. Welcome to episode number 40. How do I know when to proceed or pivot my plan? Another Math Mentoring Moment episode. This time we’ll be hanging out with Michael Rubin from northern California. Let’s do this.

Jon Orr: The conversation we have here with Michael drops a ton of great ideas, thoughts, and takeaways for your classroom. He also mentions how transformative his experience in our six-module online workshop was for pushing his math content knowledge and pedagogical practice forward. Are you ready to do a deep dive with us like Michael did this last fall?

Kyle Pearce: We run the online workshop twice a year. Once in the spring and once in the fall. So fall 2019 registration will be opening on Friday, September 13th and closing on Friday September 27th. 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 through 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 closes you can still head to makemathmoments.com/onlineworkshop to join the waiting list in order to get notified for your next opportunity to participate.

Kyle Pearce: That is, again, makemathmoments.com/onlineworkshop.

Jon Orr: And now, here’s our chat with Michael.

Kyle Pearce: Michael, welcome to the Making Math Moments That Matter podcast. We are so excited to have you with us today. And actually we can see you, which typically we can’t see guests. But today we actually have Michael on the screen with us. I’m sporting a tee-shirt because I just got out of baseball practice with my son. And Jon is rocking the basement. We’ll probably see his cat walk by in the next couple minutes. Usually your cat is kind of hanging out in there. How are you doing today Michael?

Michael Rubin: I’m doing great. I’m excited to be here. I did my hair. I knew I was going to be on video so I did that and my beard.

Jon Orr: Awesome job. Can you just let our listeners know just a little bit about yourself? Where you’re from, what’s your teaching role, and maybe a little bit of background on why you became a math teacher.

Michael Rubin: Yeah, so I’m in northern California. I teach in the city south of San Francisco. So I’ve been doing that since 2007 now. I guess about myself I’ll just start from the beginning. It’s a downer to start with, but it’s actually one of the best things that has ever happened to me, and I think that will come out too. It has determined a lot of what has happened in my life. I was actually born with a degenerative heart disease. And I think the first thing where that helped me is I grew up always knowing that I wanted to do something purposeful. I wanted to help people.

Michael Rubin: When I was really little I wanted to be a lawyer, and I decided not to do that. That probably was one of the first things that led me to teaching because I felt like I could really make an impact on the future there. I also struggled a lot in elementary school and even some of middle school. I didn’t start actually liking math or getting math until high school.

Michael Rubin: Once that started happening I noticed that I could really understand where other people struggled with things. So I was tutoring other kids and helping them out. And I think that also led me to this idea of wanting to become a teacher thinking that that was something where I could help.

Michael Rubin: So I got my teaching credential, but the heart disease had progressed to a point where I couldn’t teach full-time. Which, at the time I thought wasn’t a good thing because it was harder to get a job teaching part-time. So I basically took the job that I could get, which now I’m really grateful for. I ended up teaching three classes of English language learners. And it’s also the school where I probably would not have been working had I been able to work full-time, where I met my wife. So that was a blessing.

Michael Rubin: So I taught from about 2007 to 2016 when I just couldn’t. The stamina issues had made it so that I couldn’t give my kids what I wanted. I had a really supportive district so I ended up becoming a math TOSA. They’re working primarily with our special education teachers trying to bring the RSV classes up to speed with Common Core. Particularly with some of the math practices and how to implement them in their classes. And I had a heart transplant this last September. So I am planning on going back to teaching in August.

Jon Orr: Wow, congratulations.

Kyle Pearce: Yeah, congrats.

Jon Orr: That’s quite the journey.

Michael Rubin: I wrote it down because I needed to read you it.

Jon Orr: I’m sure. I don’t even know what you say to that. You literally had a heart transplant. I have to know, how is everything with that? Is everything as great as it could possibly be? How are you feeling? Are you feeling more energetic?

Michael Rubin: Oh yeah. Absolutely. I’m feeling great. If you’re going to have a degenerative heart disease I couldn’t recommend it enough.

Kyle Pearce: Yeah, just do a little quick swap there. Start fresh.

Jon Orr: Right. That’s amazing.

Kyle Pearce: That is amazing. You think about… I’m big into energy and the universe [inaudible 00:07:10] I believe in it, and I don’t necessarily always do what’s best for myself in terms of thinking positive. I try to. But when you talk about how things have turned out for you, like when you said you met your wife. Was it at the school? Is she a teacher at the school?

Michael Rubin: Yeah.

Kyle Pearce: So you met your wife at the school that you basically would have only been at if you had this condition, right?

Michael Rubin: Yeah, because actually I was teaching at another… sorry to interrupt you… I was teaching at another school in the district and they had position cuts. I had actually been there before some other teachers, but because I was part-time I wasn’t tenured yet. So it forced me to look somewhere else. I went to another school in the same district. If I weren’t working part-time I would have had tenure, I wouldn’t have left that school. I probably wouldn’t have met my wife.

Kyle Pearce: Holy smokes. It’s amazing the small little choices we think are insignificant that become huge and life changing for us I’ve got a few of those myself, but nothing compared to yours for sure.

Jon Orr: Yeah, absolutely. Again, it’s like when you hear a story like that it’s like everybody has something in their life where things happened and everybody says things happen for a reason. And again, the little choices we make really have a big impact over the situations we’re in. So the idea is that you’re going to be coming back in August. Sounds like you have an early start compared to us Ontario boys over here where we get to cruise into September past the labor day weekend. You’re coming in early August being in northern California.

Michael Rubin: Yeah, but we got out in late May.

Jon Orr: So you’ve already been having a little bit of fun anyways. [crosstalk 00:08:51] Well, we want to know what is your most memorable moment from math class? When you think back to math class, what is that moment that sort of pops into your mind. Go ahead and share that with the audience and let us into a little bit of your past Okay through 12 math experience.

Michael Rubin: I’ve listened to a lot of your podcast and every time you ask that question I ask myself that same question. I try to come up with something new every time. I’ve settled on one. Because I thought if I’m ever on the podcast I’ve got to have one. I like this one because it’s both good and bad. People talk about good experiences they’ve had or bad experiences they’ve had. Like most of us, I had a very traditional educational experience. The 20 minutes going over homework. The 20 minutes new material being presented. And then 20 minutes doing homework. I tended to excel in that a little bit as one of those kids that liked math.

Michael Rubin: We had a sub one day in the 11th grade. The students wanted to go over the homework. But the sub was a health teacher and hadn’t done math in a long time, especially not junior-level math. So I just raised my hand and said, “I’ll go over the homework.” I ended up getting to the front of the board, explaining the problems, answering questions and doing all that.

Michael Rubin: In a good way that validated my ideas of wanting to become a teacher. I felt comfortable up in front of other people. I felt comfortable with the math. I felt really good that I was helping some people. But on the other hand I think that the negative part of that experience was it reinforced for me the idea that being a teacher was this transference of knowledge. I’ve got the solution pathway and the methods in my head. And being a teacher is just explaining it in such a way that it will fit neatly in your head. I carried that with me for a long time.

Michael Rubin: So I think if I had to settle on one memorable math moment that one always stuck with me.

Jon Orr: That’s interesting. I think if you think back to it now… maybe you might agree. Maybe not… I’m picturing you as this student up there doing the teaching. I’m wondering whether a moment like that was the moment where you probably learned the most ever in one spot in math class. If you really think about this idea that you got to go up there and essentially have to think things through and have to make sense of it enough to try to help others make sense of it. It’s that whole meta cognition piece and teaching others. That sort of reciprocal teaching idea there.

Michael Rubin: Yeah. I noticed that also… I’m sure any teacher listening to this notices the same thing… when you have to explain something or try to tool into somebody else’s head to figure out what they’re struggling with and how they’re going to make that conceptual connection, figure out that relationship or whatever it is, you really need to understand it and make sense of it yourself. I know you guys talk a lot about that, knowing the curriculum, and the connections.

Jon Orr: It almost makes me think now that I have to be a professional in psychology. Like you just said, you have to imagine, what is the best way that this can fit interest heir brain. It’s like I have to understand how that kid’s brain is going to work. But also, what is all around that kid that is impeding on his ability to learn? There’s so much that we have to know as our educators. I’m imagining you as the student standing up there and all of a sudden that hits you. All of these kids are now looking at me and that kid is doing that over there, and that kid is doing that over there, and I’m trying to help them with their homework. And it hits you like a ton of bricks like I’m sure it hits so many brand new teachers, that there is so much more than just knowing the content and also knowing good strategies to teach math.

Kyle Pearce: And seeing it your way, right? You see it a certain way typically. It’s like what if somebody else saw it some other way? And you’re like, oh goodness. I never thought about that, right?

Jon Orr: Right. It’s just like the socio-economic around that. It’s so much. And lots to always learn every single day. Michael, do you mind if we dive a little deeper into you? Do you mind sharing… let’s think about your teaching now… a recent success in your math class. I know that you’ve been off for a little bit. So if you think back to some of your teaching experiences what would be a recent success in your math class? Do you mind sharing that with us?

Michael Rubin: Yeah, I tend to think of the last year that I taught I had one class. It was for English language learners. I had been a very traditional teacher. And a social studies teacher friend of mine said, “Hey, have you seen this guy Dan Meyer? He’s got this Ted Talk like math class needs a make over?” And I’m like no, what’s that. So I went ahead and I watched it and I was like, “Oh my goodness this is so awesome. I have to do this in my class.” I’m an overachiever when it comes to things like that.

Michael Rubin: So I was like I’m not using anybody else’s. I’m going to make my own. I spent 18 hours coming up with my own videos and learning how to do all that stuff. I remember coming up with two. One was the classic stacking cup systems of equations. Like you got this cup with different lip heights or whatever. When will they be the same? And tother one was the toothpick staircase where you could model the linear growth versus a quadratic depending on whether you were looking at the perimeter, the number of sticks, or the number of cubes or whatever.

Michael Rubin: And I had some really cool success with that. I had kids that… just tough to work with. They didn’t want to be there. They weren’t engaged. They didn’t like the way that math was taught. And all of a sudden… I had this one student who I struggled with building a connection with this student, trying to get her to work a little bit harder. I gave her this task… we were on a rotational schedule at the time and it just fell before lunch. I was getting ready to go to lunch after class and she wouldn’t leave. She just wouldn’t leave. She’s like, “I have to finish this problem.” She wouldn’t leave. That engagement was so cool to me. I was like, “I got to keep doing this.” But it was so overwhelming because it was taking me 18 hours to make these lessons. That was probably one of the best successes that motivated me going into the TOSA role, the teacher on special assignment role, where I was thinking I want to be able to do more of that.

Kyle Pearce: Sure. I hear a lot of Jon and I in that scenario. I remember the same thing. And I remember him saying it. You’ve got this technology in your pocket. Go do something with it. I remember thinking, “Oh man, I’m going to do everything this way.” I spent a lot of time early on. I think it was time well so spent. I would definitely advocate that people engage in that learning.

Kyle Pearce: But at the same time it might not be sustainable if it’s only me creating it. Luckily we had this amazing community online of all of these resources and tasks that are out there to at least cut down or give you a really good start on where you might go with that. So that’s pretty awesome.

Kyle Pearce: So it sounds like you were kind of addicted like us in terms of when you see the light in, if it’s that one student’s eye, that one girl that you’re referencing. Really tough to engage early on, weren’t too sure how am I going to first of all make a connection, help this student to want to learn this stuff. Let’s be honest, some people say we’ve got to make the math something kids can use in their everyday life.

Kyle Pearce: I’m going to argue that’s not what kids are striving for. They’re not like, “I want this so it will help me 10 years from now.” They want thought-provoking questions, they want curiosity, and they want something that just draws them in. It sounds like you were able to do that. Now I’m wondering, could we flip things a little bit and start chatting a little bit more about what are some parts of that… it sounds like you had a great start… were there any road bumps along the way that you wouldn’t mind… I see you shaking your head yes. We’ve all had those, right? I’m wondering, can you help highlight some of those pieces? Obviously you’re more than welcome to talk about other successes as well. We want this to be super asset-based. But then we also want to look at where are those road bumps. And people listening at home I’m sure are going to be able to relate to them because Jon and I will. So where are those struggles? Where are we getting pitched up on?

Michael Rubin: I think I can keep this asset-based as well even though there’s struggles. It’s a lot of the stuff you guys talked about. I had several main concerns that you talked a lot about. Is that while that one student wouldn’t leave for lunch I had another student who was the student that typically came into class, paid attention, figured out the procedures, solved the problems. And this student had a meltdown. I gave that same task and they didn’t know what to do. They just had this meltdown and I didn’t have any guidance to know what do I do about that. I didn’t know anything about grouping strategies, vertical non-permanent services, visibly random grouping, any of that. I was just having him work alone.

Michael Rubin: I read somewhere that you’re supposed to ask questions and not just give kids the answers. So I just mumbled something and moved on. I totally didn’t help with that at all. And I’m going to make this asset-based. I guess this is going to apply to everything. It was actually in November. I had just gotten out of the hospital from the transplant and I was like, oh man I have to get back into the classroom. I’ve got to figure some of this stuff out. And just that day I think it was Jon on his blog posted the how to change a textbook problem to make it a math problem that matters. And I read that and I was like, “That’s what I need. This is it. This is it. This is what I need.”

Michael Rubin: With all the different links. It linked to the three-act task. I think it was on Tap Into Teen Minds where it talks about how guys struggled at first with making a lot of those connections. These guys know what I’m talking about. These guys are going to address the issues that I was struggling most. This idea of how to help kids and how to question when you referred to the five practices. I started reading that and I was like, yeah bingo. This would have helped so much if I had known.

Michael Rubin: I didn’t have an objective. I just started the task. I’m like this will be good to bring out this idea or that. I didn’t have an objective. I had no idea how to consolidate. I didn’t do the problem ahead of time. I was like, oh this will be cool. This is how I did it. So I’m walking around and kids are doing some stuff.

Kyle Pearce: Yeah. You’re like, “I know the formula, right?” I’m like, “I got this. I can get an answer.” Yeah.

Michael Rubin: And some kids got right to the formula. Some kids were doing something. I’m like, “Oh, that’s really interesting what you’re doing.” I hadn’t pre-planned any of it. I hadn’t anticipated what they were going to do. And I certainly didn’t do anything with that consolidation piece where it was like, okay we’re going to present. So I just left it. The bell rang and we leave. And it’s like did anybody learn anything. That was something you guys have helped me with immensely in the resources that you’ve shared. And that also gets to the other thing. This idea of there’s this rigorous math content.

Michael Rubin: I’m in a district. We’ve got math we’ve got to teach. They say, “Hey, what are you teaching? How do you know you’re teaching it? How are you getting to it?” There was no guidance until I stumbled upon your blogs and stuff about how do you teach through tasks to teach the math? It’s super engaging, super cool. But how do you actually push that thinking forward? I guess that’s the last thing. It’s the how guidance on how to teach the curriculum. It’s like okay I’ve got this cool task and I’ve inserted it because it’s cool. But I don’t know what exactly to do long-term and how that all fits.

Michael Rubin: I recently took the Make Math Moments That Matter workshop, and you guys talked a lot about the learning progressions. I’m like, oh that’s how you do it. And I always struggled with the tasks. What happens when the kid finishes super early and you’ve got one kid that is just making, one group that’s just starting to make it through. It’s like oh yeah, if you know the progression you can have it prepared. Yeah, they’re doing some sort of linear equation problems. Throw a system at them. Now I’ve got this plan, oh I know where I’m going so I can push the kids that can that way while I help other kids that are struggling a little bit more.

Kyle Pearce: I’m so happy that you shared some of that background piece of the struggles, because again, the reason we shared those resources and those tools, and the reason why we set things up in the online workshop in the way we have is exactly like you said. We were those eager guys out there like, “Boom. We found the solution to making kids not hate math class so much.” And we always give the caveat. We’re not saying that it’s going to make every single student suddenly be like… you know, like everything is perfect. But to see that change in students leaning in and just noticing kids when the bell would go and they’d say, “Wow, that was quick.” That was a huge compliment to me. I didn’t need them to say this was my favorite class. But for them to not have already been having their books packed up ready to go five minutes ahead of time, to me that was a huge, huge step in the right direction.

Kyle Pearce: But like you said, we walked out and went, I know they’re engaged but I don’t know who knows what. Good colleague of mine, good friend of mine Yvette Leeman always says, “I don’t know who heard what. And I don’t know what this means to them.” And I didn’t know where to send them next. So I’m so happy that you hit on this idea of the progression.

Kyle Pearce: For those listening who might not be super confident in that area of the progression of how different concepts build on one another, that’s okay. It’s just this idea that if you know it exists and people have different perspectives as well on how it works. It’s just knowing it’s there and trying to dig deeper. I think we’ll forever be learning more and more about those nuances in between. But it’s just about getting started, which is awesome. So it sounds like you had found some resources. And it sounds like those helped give you something to bite into. So how were you feeling in that regard when you think about how to… you talked about a lot of things there, right? You talked about The Five Practices, which is a book we talk about all the time and is a super easy read. Which is great. Someone like yourself went and grabbed it and it probably blew your mind, right? When you go, “Oh.” It seems so obvious after. You’re like, “How didn’t I know that? I never realized it. I never put it together.”

Kyle Pearce: You’re talking about some of those other pieces like progression. So how do you feel about that developing in your classroom? Maybe just give us an update on what that might look like or sound like in your class. Or at least what you hope to do in your class when you get back to business in August.

Michael Rubin: Yeah. Part of my role as a TOSA, one of the things I was doing was working on writing some sort of accommodations. I was using… are you familiar with illustrative mathematics, the six through eight curriculum? I was accommodating some of those resources for our special education RSP teachers. I ended up using a lot of those. I couldn’t give the three-act task because I feel like you do need to know a lot about three-act tasks. But I implemented a lot of those ideas into the progression of the lessons and that’s how I accommodated them. I’ve heard a lot through your blog or your podcast about lowering the threshold and raising the ceiling. It was always like, oh great, yeah, I want to lower that threshold, but how do I do it? And how do I increase that engagement? Speaking of those mind-blowing moments, your curiosity path was for sure the most mind-blowing moment I think in my career.

Michael Rubin: If I can tell somebody one thing… especially working with English language learners and special education students, so many of them see those numbers or they see that context and they’re like, “Oh, I can’t do this. Somebody else is going to do it and I’ll just copy off them.” This idea of making it visual. Withholding information. I just started incorporating that in the lessons and I would write some directions to the teachers of how they might go ahead and do that. Having them notice and wonder. And then estimate, anticipate some of those ideas.

Michael Rubin: So as I plan on going back to the classroom I’m feeling pretty comfortable with a lot of that. Granted, I’m also feeling comfortable knowing that it’s probably going to be a disaster some days as I try to implement teaching methods that students probably, I’m assuming, haven’t been exposed to before. But I think it will really help. Especially since I’m going to be teaching three course one courses for English language learners. So it’s a mix of beginning algebra and geometry. I’m expected to teach the same exact content as everyone else to some kids who… my wife is actually the ELD one teacher, and I went and met some of my students I’m going to have next year and they’re at the, “Hi, my name is…” stage. So it’s like how do we get them involved in the task? I think that sparking curiosity, withholding some stuff, making it visual, providing the language supports around that and that structure will really help get kids thinking about the context, understanding the problem so it’s not so overwhelming for them. And then they can use those visuals.

Michael Rubin: Again… I know I keep referring back to this… but in the workshop you talked like be more like Prince. The concrete representational abstract model. That idea of making it visual and having them understand some of those ideas, and have an idea of what they’re doing before they engage with math might be really helpful. So I’m feeling really confident with that. Probably more confident than I should be.

Kyle Pearce: I just want to comment and say just based on what you’ve said… first of all, I’m looking at Jon right now as well and I can see it in his eyes… we feel so happy to hear that you took that content. It’s so easy to sign up for something, go in… it’s like we all have lives. Life is busy and you have things going on. It’s really easy to go like, “Wow, I really want to learn. I really want to do these things.” But just in hearing what you’re saying. And the fact that you’ve been out of the classroom through your recovery and then getting ready for the classroom coming up after summer in August and how you’re speaking to these points so fluidly really tells me that you’ve done a huge amount of reflection. And I think also just for you to be aware at least conscious of this idea like okay, I think it all makes sense but we know how that goes, right? I think I very it all worked out every day I present my workshop or every time I go into a classroom. And every single day I walk out of there and I go, “Wow, there was always three of our things I was not ready for that.”

Kyle Pearce: And in many ways it’s like you can’t always be ready. But the fact that you’re aware. Like your awareness is so high. And I just want to tip my cap a little bit to just the fact that you’ve done that thinking. You’ve done a ton of learning clearly. I think that’s going to put you in such a great position so that when that curve ball does come… because it’s going to. You don’t know what pitch it’s going to be, but it’s coming… at least you can be ready. Sometimes it’s going to buckle you at the knees and you’re just going to go, “That one got me.” But you can feel good about the fact that you have done so much work ahead of time that I feel like you’ll be able to get that next pitch. I didn’t want to interrupt but I do want to make sure you have that feather in your cap, because clearly you’ve don’t a lot of serious thinking here.

Michael Rubin: Yeah. One thing I’d like to say too… I talked to my colleagues a little bit about this. I got two of them to do the workshop with me. We formed a little study group every week and meet and talk about the videos. I feel like the more I do it the more I see how it’s done. I was tutoring the son of a cousin the other day in geometry. He was doing two-column proofs. It’s not the most engaging topic. I was looking at it like, how would I use the curiosity path here? I started seeing it had this visual. It was this simple little proof. But it had this visual. And I was like, oh I’d use Keynote and I’d make it visual and I’d have this piece moving back and forth. And I’d do a notice and wonder and value that student voice. You start seeing these things.

Kyle Pearce: Do you feel like when you do some of that work… for me I feel like you can almost see inside the kid’s brain in a way. When you ask them and you just really listen. And the hardest part I think is the listening. It’s so easy to miss or not to ask the right question. There’s tons of time after a class I come out and I’m like, I should have asked this because I still didn’t get what I was hoping to. But when you anticipate something and it does work out there’s nothing better. There’s not a better feeling in the world when you’re like, holy smokes that actually worked. You start to feel like you’re getting into some of their heads. When it doesn’t that’s when you go, oh man I have no idea where that came from but I’m going to have to think more about that. Super cool.

Jon Orr: Yeah. I haven’t said much in the last few minutes. And the reason is I’ve been hung up, Michael, on one of the first questions you said. I would love to go back to this. When you said you had that student that wouldn’t leave the class. But then you had that high flyer student that was just in… I don’t know if anyone else listening right now still wants to know how we’d handle that. That kid who’s like, “I’m really great. I very been successful at math and all of a sudden what are you doing tome? I’m really frustrated right now.” You said at that time you didn’t know what to do. I’m wondering now after you’ve gone through some learning, you’ve had a little bit more thought and reflection what would you say to this student? It’s been on my mind since you said it. So I would love for us to just chat a little bit about that and then we’ll dive into some questions that you have and struggles that you have for ongoing stuff. But I’d love to get your take what you would do or say to this student now.

Michael Rubin: Yeah, and I do want to be cognizant of your time as well because once I get talking I tend to run on.

Jon Orr: Oh no worries.

Michael Rubin: Okay. I didn’t know. You’re three hours ahead of me, and I’m on summer break. Man, I love that question because you’re putting me on the spot so much. I would like to think… I’m going to break this down into two parts. And I’m thinking on my feet here. I would like to think that knowing what I know now I wouldn’t have that kid there. I wouldn’t have a kid that didn’t have a strategy.

Michael Rubin: One because he wouldn’t be working alone, first of all. I would definitely have him working in some sort of group. I’ve already bought the white books that you recommended. I got them prepared. Figuring out where I’m going to put them in my class. So I’d have him work in groups. And I really like this part, again, the curiosity path where you have kids anticipate. That question like what information would you want to know and why? Getting kids to think about a strategy before they start. I would like to think that would support those kids a little bit more so that when they’re sitting there and they’re stuck they wouldn’t be because they have an idea of what they’re going to do.

Michael Rubin: Another thing… I don’t know if you’ve talked about this on your podcast, but I recently read a book that was almost as mind-blowing to me as the curiosity path. But I’m going to put the curiosity path at number one. It was Reprieves for Reasoning.

Jon Orr: Okay. That’s actually on my radar. I haven’t read it yet. But it is actually on a list right now for me.

Michael Rubin: If you don’t mind I’ll just briefly paraphrase what that book is about. And I don’t have it with me and I forget the author’s name because I want to plug them.

Kyle Pearce: We’ll put it interest show notes.

Michael Rubin: Okay, perfect. What it does is it looks at the ways in which mathematicians think. It does that from… I don’t know if you guys use the mathematical practices. You talk about the process standards.

Kyle Pearce: Yeah, we have similar ones.

Michael Rubin: Okay.

Jon Orr: Yeah, they’re called processes, but same idea.

Michael Rubin: Yeah, okay. This book starts with mathematical practice number one. The perseverance solving problems and making sense of them. And then it talks about practices two, seven, and eight, which are about quantitative reasoning, structural reasoning and reasoning through representation. It goes through and provides routines that kids can go through with questions they can ask themselves and strategies they can use.

Michael Rubin: So the quantitative reasoning is you look for the quantities. The quantities being a number, a value and a label. But that value could be unknown so it could be a variable and then you look at the relationships between them. A number could be a quantity or it could be a relationship. But you figure out what those are. You figure out what the relationships are and then you draw a picture that represents it. And by drawing a picture and making it visual it might give you some sort of insight into the problem.

Michael Rubin: So when I think about that kid that doesn’t have a strategy or doesn’t know what to do one of the things I’m planning for my course one class next year… I’ve actually made them objectives… are the mathematical practices where I want kids to reason quantitatively. And I’m going to explicitly teach them through these routines how to do that. So if I did get to a point where a student might struggle with that I wold hope that we had worked on some of these strategies by then because I plan on baking it into the class. If they say, “All right, if you’re not quite sure what the or do, what’s a question you could ask yourself? What’s a way you could reason quantitatively or structurally noticing how to solve the problem like this before?” Or if you made a table, how are you calculating the values in the table where you’re not just using it to guess and check, but you’re using it to develop a process and then generalize that process so you could come up with a formula or a rule or something that I would still be able to ask the question, bid be able to ask it in a way to talk about a strategy that they can use to help solve that problem.

Michael Rubin: This all sounds great in my head. We’ll see how it works.

Jon Orr: No, these are great tips for sure. They are good, actionable tips for people to think about how can I help those students. That’s a common thing. You deal with that on an every day basis. There’s always kids that are going to struggle. Even when you lower the floor very low. There’s going to be parts when that floor starts to rise. Everyone hits that frustration level. Not everyone, but a lot of kids hit that frustration level depending on the grade level and the skill level the kids have. But these are great tips to have and that you’ve given us to think about from that book.

Kyle Pearce: I really like how you frame that where you were saying hopefully students aren’t in that position because you’re creating that classroom culture in your approach. And I really think too, you mentioned it earlier, the low floor high ceiling…. by lowering the floor, and especially doing it early on. When we do it first week of school. Like you folks going back in early August, kids are back in the classroom and you really try to focus less on curriculum and more on the math practices early on in the course I think really can help students defeat that idea. If I start doing this stuff in the middle of the year then a student, especially my high flyers right? They’ve got the game of math figured out. And it’s like I don’t know how this game works and all of those are going on.

Kyle Pearce: So it sounds like from what you’re saying, I think I would agree with you. Hopefully building that culture early on students will, if they do have any issues, it’s like the stakes are really low because it’s the beginning of the school year. It’s not like exams are coming up or midterms are coming up, whatever it is. It’s like right away, right out of the gate, here’s the rules in this class. We’re going to do things a little bit different than how maybe things were done in some of your other classes.

Jon Orr: Let’s probe a little more Michael, what’s on your mind lately? What challenges are you experiencing that you want to chat with us about?

Michael Rubin: Yeah, this is probably what I really want to get into and talk to you about. As I said, I’m going to be teaching course one for sheltered students where some of them are going to have some pretty good basic interpersonal communication skills. The academic language might be a little bit harder. And some of them are going to show up four, five, six, seven months into the school year speaking no English. They’ll show up and say, “Hey, I’m here to learn math.” And I’m responsible for teaching them that.

Michael Rubin: And I had never, before the Make Math Moments That Matter workshop, I had never before heard of spiraling. It was the last word to be released. I saw this word spiraling. As the weeks were going on I was like, “What’s that about? What’s that about? I want to know that’s about.” And I finally got to it and I was like, why don’t we all do this? I don’t understand. This makes so much sense. And I’ve actually been exposed to a curriculum that did it. It was CPM. I don’t know if you know CPM.

Jon Orr: Oh yeah.

Kyle Pearce: I’ve heard of it.

Michael Rubin: But I didn’t know what they were doing. I just knew that they bounced back and forth. I was looking at that and saying, not only do I think this would be really good for the reasons that you said. It causes students to have to remember what they’re doing. It gives them more opportunities to access the materials. It keeps them from getting bored because it’s not like, “Hey, we’re doing unit three on this topic and we’re going to do that for the next four weeks.” How are you going to introduce a task and surprise them when they know what it’s about?

Michael Rubin: But also, the challenge that I’m worried about the most is that… my wife will get new students in March and we finish in May. If we’ve already taught, we’re just in the geometry part and we’ve already taught all the algebra how am I serving that kid if I don’t go back and do some of that? So when I saw this concept of spiraling I’m like, I’m doing that. But it’s a little bit difficult to do.

Kyle Pearce: Do you mind telling us a little bit more? I tend to agree with you that it is a challenging thing.

Jon Orr: It’s a big change.

Kyle Pearce: So what about it is feeling like a big thing here?

Jon Orr: Sorry to cut you off. Maybe in your own words explain what you think it is. For people who are also just hearing it right now.

Kyle Pearce: Yeah, that’s actually a great question.

Michael Rubin: What I think spiraling is or my process?

Jon Orr: Yeah, spiraling.

Michael Rubin: So spiraling is when normally as math teachers if you look at a textbook you’ve got chapters. In my school’s textbook we’ve got chapter one, chapter two, chapter three. Every chapter covers a different topic. It cycles through. So even though we’re a course one integrated school, we’re on an integrated pathway, we still teach it like the first semester is algebra and the second semester is geometry. So we’re not actually integrating any of the ideas. And the idea of spiraling… I’m just regurgitating what you guys have said. So if I say something wrong feel free to correct me. But the idea is that you break it down into chunks. So you might spend three days on an algebraic topic. And you might spend three days on a geometry topic. If there’s any way to connect them that’s great because you’re now doing these things together. Then you might go to statistics or whatever. And then you might come back to algebra again.

Michael Rubin: So you’re spending several days on that. Every time you do that you go a little bit deeper. The first time… I’ll just… because I’ve been doing it with course one and these are some things I have questions about as well… in my mind, the first time you do that you’re going to present students with a problem where they have to write a linear expression in one variable. The second time you might have them write a linear expression and one variable that maybe is set up in such a way where you could use the distributive property to simplify it. So every time you go through it, it’s the same concept but they’re just going a little bit deeper so they have to remember and recall what they’ve learned. But then you’re just taking it to that next step.

Michael Rubin: The benefits to that, which I’ve touched on a little bit and you’ve touched on a lot, is that one it forces students to remember. I know my students have always really struggled with retention. So something I taught six weeks before you provide them a problem on a standardized test or benchmark assessment and they’re like, “I don’t remember doing this,” because we just taught it once. That idea that teaching by units gives us a false sense of mastery where hey, they did this test really well so now they know it. It’s like how do you know unless they can do it again in six weeks? Forcing them to recall it helps with that.

Michael Rubin: There’s also the idea of boredom, right? You know it’s coming. I’ve had the experience where I’ve gotten into a topic and students are really struggling with it and they just know for the next two weeks math class is going to suck because they hate this particular topic or they struggle with it. That’s what we’re doing for the next two weeks. And if I feel it’s important we’re going to be doing it for three weeks or four weeks. And moving things along it just meyers everything down. It also ruins the… if you teach by chapters it ruins the surprise in the same way. Especially if you’re trying to teach through task.

Michael Rubin: It’s like hey, here’s this interesting task. What do you notice and wonder? Well, we’re doing linear relations right now so you want me to say something about linear relations. Let me figure something like that. There’s and surprise. No mystery there. All these are reasons I’m like, I have to do this.

Kyle Pearce: I love how you’ve described it. I feel like you’ve done a really good job of summarizing. I’m going to back up to even your example, because your example was… and you were very explicit about it just an example of like… because you said, maybe three days of this and maybe three days of that. One of the biggest things that people ask us is, “How many days should you spend on something?” The big answer is it depends. It could be three. It could be five. That’s more or less for anyone who is listening or watching this, just to make sure that they know, “Okay, there’s no one right way to do it,” but how you framed that out I think is very clear.

Kyle Pearce: And I also think, too. Even just this idea of coming back to these ideas and for going deeper with them it just forces you… and you hit on it… this idea that it’s forcing you to go back and really utilize and push your brain to put things into long-term memory versus it being really fresh and getting that illusion that you know it when really it’s just familiar because it’s from the past little while.

Kyle Pearce: Super cool. Now that you’ve shared that we’re wondering, as you do this… so there’s a lot to that, clearly, based on what you said. Where are some areas that you’re particularly seeing as little hurdles here that maybe we can hash out. And we’ll keep on probing a little bit to try to get deeper and deeper and really see.

Michael Rubin: Yeah. Is it okay if I give you my general process? You guys have done this and I don’t know if my general process is just going to completely not work.

Kyle Pearce: Yeah.

Michael Rubin: I actually was fortunate enough to be on my district’s scope and sequence team this year. So I was part of figuring out what are the essential topics and standards that we want to cover. I took that and I used that to organize by the topics and the big ideas. Just for some reference, you’ve interviewed Joe Bowler before with the Mathematical Mindsets. She talks a lot… I’m plugging everybody here… she talks a lot in her book, Mathematical Mindsets, about how you need to value in your assessment practices what you value otherwise the kids won’t value it. You need to make it worth something.

Michael Rubin: So I organized by the topics not big ideas. My first objective is… and I’ve got a rubric for it… I can work collaboratively with others. And then the next three are pretty much straight out of that Routines for Reasoning book where I can reason quantitatively…. I’m just paraphrasing here… I can reason quantitatively. I can reason with structure and use structure to solve problems. And I can reason through repetition. And then I get into the contents standards. What I was trying to do… because I’ve never really done this before, but I’m planning on doing some standards-based grading. Really just doing standards-based grading.

Michael Rubin: So I didn’t want to have 100 objectives that I was grading kids on. I was thinking of having a main concept idea. So just as people are listening, if they’re not looking at it, one of my topics is algebra single variable. And the objective is I can write expressions and equations and one variable and use them to solve problems. So you’ve got all sorts of different topics within that. You’ve got simple single-variable problems, you’ve got the distributive property, you’ve got equations with variables on both sides. You can have all these different levels of complexity. And my idea is as I spiral deeper into these topics and they meet some of these sub objectives they’re gaining credit towards having mastery on this overall larger objective. So that way I’m really on grading them on these 13 objectives. I organize it by that.

Michael Rubin: I wrote a bunch of I can statements and then I listened to one of your recent podcasts where I’d never heard of the know, understand, do the KUD model before. I’m like, perfect. So I quickly went through and started… it’s quite fun to do actually to figure out where’s the understanding, where do they just have to know, and what are the skills? And I thought it was important. First I was like I’ll do that next year or whatever. But I was looking at that and thinking that might help me really break it down. And that will lead me to one of my questions as well with spiraling. This will help me break it down and figure it out.

Michael Rubin: And then my plan is to go through and search for, before I even break anything down, search for tasks. I feel like this is going to be the meat of it. I’ve got a list… if you look at it… I think it’s on page three of three-act math resources and inquiry tasks and other task resources. I’m planning on looking at two websites. The three-act math tasks, the task search, Joanna Roe. I think you guys are familiar with him on Twitter. He’s got this great-

Kyle Pearce: Actually, we’re going to be chatting with him this weekend I think. He’s coming on the podcast.

Michael Rubin: He’s been really inspirational as well. I’ve got the mathematics assessment project, illustrated mathematics, enrich. And I’m just going to go through and start looking at tasks and figuring out what are the main ideas and concepts that come out of these tasks and where do they fit?

Jon Orr: I was just going to jump in here because that’s exactly when Kyle and I began this process the first time, you’ve done way more thinking about it than we have.

Kyle Pearce: We just rolled up the sleeves and went. We were like, “Let’s just try this.”

Michael Rubin: That’s patently not true because you guys didn’t have you guys. I was able to take a lot of those ideas. It’s pushed me.

Jon Orr: What we did is what you just mentioned. We taught through task first. I’ve said this before, I quote our good friend Hal Overwick who was also, we learned, he was spiraling way before us, teacher from Ottawa, Ontario. He’s always said teach through the task first and uncover the curriculum instead of curriculum. He was always a proponent of find a great task that you can bring out the math that you want to talk about. That’s how we started with our courses. It’s like what are the tasks we can bring. If we want to do, what you said, solve equations or using algebra to work with equations what is a task that we can use to bring that idea forward? And then we can focus on that idea. So that’s what we did. We just went through tasks and then we consolidated together on those tasks some. Kyle what tasks are you doing for that expectation? What task am I doing? We made a big list of that. That’s how we started to begin and everything else we learned along the way. You’re starting way farther than we did, but you’re also starting to think about tasks.

Michael Rubin: No, I’m just copying what you did. I participated in a lot of your stuff and I’m very grateful for the work you guys have done in general. You led me to all these different areas and ideas and makes these things seem not so intimating.

Jon Orr: That’s awesome. That’s the intent, right? I’m not even saying maybe there were people out there. We just didn’t know where to find them, right? [crosstalk 00:48:46] and we had access to a bunch of amazing tasks, which without that we would probably just be slugging away doing it the same way we were doing it for so long. Let’s dive a little deeper. What else? What else is… you’re seeing that hurdle. I love this idea of starting with tasks and you’re using that to form how this might be organized. What do you see as you picture that, what do you see as something that might be a bit of a challenge for you?

Michael Rubin: I’ve got several. As I’ve gone through and done this these questions are just popping up. One of them that I’ve thought about a lot is, okay, I’ve got these really nine different things that I’ll be spiraling. Then the collaboration and the reasoning skills will be insert. My general plan, just to give some perspective of how I plan to do this, is the first four, which is the collaboration and the reasoning skills, I’m planning on interspersing throughout so none of those will get their own chunk. It will be okay, we’re dealing with objective five or whatever and I want to focus on quantitative reasoning so I’m going to go through that routine with some sort of problem involving objective five. So as I think about going through and doing this one of the questions that comes up I still have nine objectives. So if I hi one, how long do I wait? I go through all nine and then come back maybe a month later? That seems a little excessive. What would be your recommendation for how long you wait before you return to go deeper into a single progression?

Jon Orr: Yeah, I think there’s a couple ways you can look at this and there’s also a couple ways you can organize your content and what you’re going to talk about. I’ve heard it done a few different ways. One way, and the way we did it, is not something I’d recommend. But all we did is load our tasks… We always say this, say Kyle, we loaded our tasks in a shotgun and then we just shot it at the year. We’d spend a day or two on this one. Then we’d switch gears and we had no rhyme or reason to switch to another one. It was like when that task was done and we focused on one learning goal or two learning goals for that task when that was done we just grabbed the next one that was in the line. The line was random almost. We did the next one which was another learning goal. I think you can be more organized. It sounds like you’re working to be more organized than we were.

Kyle Pearce: Jon, can I interrupt you?

Jon Orr: Sure.

Kyle Pearce: I notice from my class… and I think you’d agree, but feel free if I’m off… while we didn’t have that intentionality, and Jon hitting on this idea of looking back on it it would have been way better had we had more intentionality on why we put certain tasks in certain spots the one thing that might be a bit disappointing for ourselves to reflect on is the idea that it’s not like student achievement went down either. So I’m like if we did this with no intention and actually student engagement was higher, but student achievement wasn’t affected that’s sad to me because I’m like if we just randomly did it and there was no rhyme or reason to it and we got essentially the same outcomes we typically get when we did it in units that makes me believe, oh my gosh how many years were we doing units where I thought I was being intentional but clearly it wasn’t having a lasting affect, that retention. Sorry Jon, I just wanted to make that clear. It wasn’t like it was a disaster happening that way. It’s just it could have gone so much better.

Jon Orr: No, no, no. It wasn’t a disaster because I think one thing that was a really good benefit of that… because people would say are your kids learning the math better this way? And at that time we could say like you just said, they’re more engaged. But I always would say that my students were better problem solvers after. It was to the effect that, like you said Michael, are we really problem solving if in the two-week span we’re on linear relations every kid knows that the next day the problem is going to be on linear relations? So when we scattered the problems and we taught through problems they didn’t know what to use to solve this problem. They had to think back in the brain where does this relate to what I’ve learned in the past and bring it forward, which strengthens your brain.

Jon Orr: Our students became, in my opinion, more confident problem solvers because of that. It was like they always saw unfamiliar problems to start class. And then we brought the math forward. Since they did that on a regular basis when you handed them the problem at any time and it was unfamiliar they didn’t have the whole drop pen, throw your hands up you didn’t show me how to solve this. That just didn’t happen anymore. So that was a really good benefit of us just scattering problems. I’m just going to give you one more that I’ve heard and I’ve used in a different course.

Jon Orr: You can also look at your content or your expectations and think about what are the big ideas, but also think of it this way. Mary Barasa, also from Ottawa, also an early adopter of spiraling, spiraled her course by doing everything… her first wave was everything graphically, visually. She solved systems of equations graphically and visually with counters and manipulatives. She was factoring expressions that way with manipulatives and all that linear stuff in that course was graphs first. Quadratics was graphs first.

Jon Orr: Then it was the next spiral was starting to link the graphing work that you’ve done in all these different areas to the abstract. Much like what you learned with the concrete fading model. Everything was concrete, and then we spiraled it down to be more abstract. And then finally end up with tougher problems that were very abstract, very algebraic by the end. So you can think of it that way too. And then every big unit topic goes along that pathway.

Jon Orr: We’ll always say, I guess to answer your question about how long do you spend, it does depend. And it also depends on the kids, right? It depends on whether the kids are ready to move on. When you taught to that task and brought the math out forward and then practiced that learning goal that you’re wanting to bring out, how are they doing on that? Are they ready to move onto the next one? Sometimes it’s like maybe they’re not ready, but that’s okay. You know you’re going to hit it back later. So I think usually what I would do is think how long does it take to get that task and that learning goal feeling somewhat confident for your students before you move on.

Michael Rubin: Okay. I’m inferring from that as well is maybe… one of the things I thought about is what if students are particularly struggling with a task? And I’ve noticed sometimes students that struggle they’re just like, “I just don’t want to do this anymore.” One of the things I’m hearing a lot is have that flexibility to say, “Hey, maybe I plan to come back to this in like three weeks. But they didn’t really get it and they need to move on right now where they’re at. They just can’t do this anymore, so I’m going to go into the next chunk and then maybe rearrange it so I come back to it in a week.” And maybe hit it from a different angle.

Kyle Pearce: I love that idea like that’s one option. The other option too is trying to be responsive to the student learning needs in the class and thinking, okay, well if they’re not getting it… and when we say getting it we’re talking like students in general is just not catching on. Clearly there’s something missing here. Sometimes giving yourself that time can be nice for you to now reflect on it and say, “Okay, when I come back to this I don’t want to wait until I say I’m coming back in a week and not think about it until the day before.” Between now and then you’re going, “Okay, what is it that students are actually struggling with with this topic?”

Kyle Pearce: Sometimes what we find if we dig a little deeper… what I pick to pivot to… we call it pivoting or proceeding… so I have this plan. And maybe I have to pivot the plan because I’m going to do something tomorrow that might not directly fix or take care of the current misconception or issue or challenge. But by doing that it’s something that I know is before this thing that students are struggling with. So is it possible, can I possibly set up some sort of structure in my class where I can give purposeful practice for students to work on and I can pull small groups and try to learn more about what they’re struggling with. So actually work with four or five kids individually and say, “Hey, we’re struggling with this, I’m wondering is it because of that.” What’s that concept before it that might actually be the thing hindering it?

Kyle Pearce: So for example, if I’m working with a student and they’re struggling with a certain type of division, maybe corporative division. If I just keep hammering division but what I realize if I’d back up a little bit is they’re actually really struggling with this idea of skip counting. And skip counting backwards in particular. If they’re approaching division as repeated subtraction that ability to fluidly and fluently do that could be the thing that’s holding them up here. It might not be that they don’t get the idea of division itself, they just might not have the number flexibility to do it. So depending on what that topic is, maybe what you do tomorrow might be a concept. Like maybe it’s going back to something from earlier in the year. Even though I was going to spiral back to this in a few weeks, maybe it’s time for me to go back to that place because maybe that concept itself is the thing that could push them over that little hurdle.

Michael Rubin: Right. Can I actually ask a followup question to that?

Kyle Pearce: Absolutely.

Michael Rubin: You used a word that I’ve seen a ton. And I’ve actually used it myself and I have a concept of what it is, but I’ve never actually seen it defined, and I think a lot of what you just said hinges on it as well. Saying that you’re going to have kids do purposeful practice. What do you mean by purposeful practice?

Jon Orr: Yeah, purposeful practice, it’s not like one specific thing but it’s like picking those practice problems with more intentionality than what I used to do. Which was like, this is the worksheet that I’ve always used and I’m just going to go ahead and hand that over. Or sometimes what I’d do was I had the worksheet or I had the textbook and I was like the homework questions I give out every single year are one, A-C-E. Two, B-D. This one is like A to D. But every class has a different group of students in front of me. So it’s like me actually looking at that and going, “Okay, wait a second, what do these students actually need?” And a good place to start could be this idea of as a whole group what does this whole group need? Maybe one and two aren’t helpful and necessary at all. Maybe that later question is actually way beyond what students are ready for and I can see that right now just through my formative assessment right? My observation and my conversations. So rather than being married to this idea that every year… I think as teachers we get so married to our plan because last year a kid did, we’re going to call it okay. And if I modify anything will that be the thing, the straw that breaks the camel’s back?

Jon Orr: So we don’t like to change those types of things. We’re like well, I gave this much practice every single day and things turned out this way. If I take away some of those problems or if I change some of those problems is that going to help or hurt? And I’m going to argue that if paying attention it will actually help. If I’m going to pick purposefully the practice problems I pick they don’t have to be… the reason I’m using the example of a handout or a textbook is because the one thing we don’t want people to walk away from when they listen to the podcast or when they come into the online workshop, we don’t want people to think that any type of practice is bad or that worksheets themselves are bad. I just think the way we used worksheets… and sometimes even the ways we used textbooks in the past was without purpose. And that is the part that we don’t want. We want to use those things for the… there’s lots of great stuff in textbooks. Really cool explorations. I just used to always skip them because I didn’t actually teach with intention and I didn’t realize what I was missing and the decisions I was making. That’s mine. Jon, anything else?

Jon Orr: Yeah, for sure. I think another example of purposeful practice… usually this is the one that pops into my mind when I think of it is practice that’s embedded in the task itself. So for example, if heard of open middle from Robert Kaplinski and others, you throw up an open middle problem where there’s a whole bunch of blanks in this solving equations thing, this prompt, and you’ve got to put in the numbers to make the smallest solution. Think about how much practice is going on in that problem itself just to tackle that one task. Kids are putting numbers in, they have to solve that equation. Did they get a low number? I don’t know. I won’t know until I do it again. Then again, and again and they compare to their classmates. That’s one of my go-those to explain purposeful practice because you’re practicing in that activity purposefully to solve this other thing.

Jon Orr: Another one is if you’ve used Dan Meyer’s Dandy Candies task, which is on surface area and volume of different shapes. What’s maximizing the volume given a certain amount of surface area? So there’s different shapes on the screen, you have to find the volume of each of them in order to figure this problem out which is bigger than just doing that. You have to calculate the volume of various shapes. And you have to calculate the surface area of different shapes, which is practice built into this other big idea. So that’s another example in addition to what Kyle said as to what purposeful practices could be.

Michael Rubin: Yeah, thank you for that because I was thinking more along the lines of Jon’s line of thinking. It was like instead of just giving them a page of 40 problems they need to use what they’ve learned in a context that is like they’re taking it almost a step further, applying it to a different context in a similar way but it’s not just like we’re going to do this procedure. But I really like that idea of thinking about the student needs. This intentionality of being like what’s going to be best for the student. Not necessarily just, hey this one problem is going to be great, and thinking about that. And also saying, well which kids is it going to be better for? How can I tweak it? Being intentional thinking about what your kids need. I like that idea lot.

Kyle Pearce: I was going to say when you are using a task you can actually build out some of those problems as… we call it extensions. Dan Meyer calls it sequels sometimes. Usually his sequels are still idea based in some way. But even incorporating the same context, you work so hard to develop the context that to have a series of problems that take that context… and it could even be a completely different scenario. But it’s like if you’re talking about this context you can use something like it where it’s sort of like you can leap frog from it. And someone who does that really well, I find, is Cathy Fosnot, her context for learning kits. She doesn’t like them being called kits. They’re like units of study that she used. And they’re designed for elementary so this will probably not help for your specific scenario. But she does a great job of setting the context and then she rocks that context sometimes for multiple days. And the context is just being modified a little bit.

Kyle Pearce: What I find with those tasks is you invest a lot into… I tend to do a notice and wonder even with her resources when I’m in an elementary class, even though that’s not necessarily the protocol she suggests you use, but I do. And then the next day I can go into the next part of the task and it’s like we’re into the task really quickly because we’re using all of that curiosity that built the previous day. Students feel really grounded in the context. It gives them a great leaping off point. That’s a great way to do it.

Kyle Pearce: Then also, keep in mind too, you mentioned the word procedure. And I just want to caution people listening that one of the effective teaching practices from Principles to Actions… which is a great book for anyone thinking about changing math practice in a system in a school. It’s by NCTM. One of those math practices, or those effective teaching practices I should say, is building procedural fluency from conceptual understanding.

Kyle Pearce: I think early on in my journey it was sort of like… I didn’t mean to do this, but I think I almost turned down the idea of a procedure, like it’s not allowed. But now what I realize is that the ultimate goal is that we build this conceptual understanding. We make sense of the mass so that I can get to a place where I can use things with fluency. And I want to build the procedures. But I want to understand them. So that’s key.

Kyle Pearce: Sometimes some of that practice might look like something that’s more traditional. That’s okay too. We just want to make sure kids are at this place where when we get that far off practice… like the practice I initially described… we don’t want them feeling like, first of all, it’s way too much and pointless. We don’t want them to feel like it’s not accessible to them. If they have a conceptual understanding, then those type of problems to help them build that procedural fluency will develop over time and it won’t be as boring or overwhelming or inaccessible like it may have been in the past when I was in the classroom and some of my friends weren’t able to do the homework because they just didn’t have the conceptual understanding to get themselves out of a jam.

Michael Rubin: I like that for… One of them is with English learners particularly, understanding the context can be one of the hardest things. Understanding the problems. I like that idea with that practice, this extension question, using the same context because then they’re there. Okay, I understand this context. I don’t have to work that out.i can focus more on the math.

Michael Rubin: And the other part… and I think the reason why I tend to… I hear what you’re saying and I need to keep reminding myself, procedure isn’t bad. I think I tend to make that connection though because I know myself as a teacher. When teaching students that don’t have fluency with the language it’s so easy to say, well they struggle with the language so I’m getting rid of all of it and I’m just going to that procedure because if I do the procedure the kids can copy what I’d and they can feel successful and I can feel successful. But in the long run they’re not learning anything. So to keep in mind that purposeful practice, it can and should include some of that procedural ideas. But after that context and those concepts have been developed.

Jon Orr: One of the goals is to try to get to that procedure. The procedures have been invented to make our life easier. If we can bring that to your kids’ eyes to say, this procedure we’ve now funneled ourselves into, we just do it together and then it happens later it would just make an eye open. They know exactly where the procedure comes from. Which probably our parents and us never knew. But that would be the goal of my class is I do want to end up having those students leave my class with those same procedures. It’s just how I develop those different.

Kyle Pearce: I think it’s like we don’t want to rush it. We talk about not rushing the algorithm in the workshop quite a bit. But then also the other piece is we don’t want them to only walk away with procedure, right? And we don’t want them to only walk away with conceptual understanding. When we look at the five proficiencies, which are outlined by the National Research Council and come up in the Adding It Up document or the Adding It Up book, chapter four is all about those five proficiencies. And two of those proficiencies, one is conceptual understanding and then one of the others is procedural fluency.

Kyle Pearce: So it’s this idea of it’s not an either or game. We do want both. But the research does tell us that if we rush the procedure then students tend to not want to know how it works because it’s like who would? If I know the shortcut I’m going to do the shortcut. But it’s like then we miss all the math and all the math thinking. And then we also miss the opportunity to bail ourselves out when things are getting tough. So as the complexity of the tasks get more significant and things get harder it’s like how do I break this task into something more manageable for myself if I don’t understand how the procedure works in the first place? You can go back to things like area of a trapezoid, right?

Kyle Pearce: For a lot of kids that is a straight procedure because they never understood or never realized, hey that’s just two triangles put together. Wait, I don’t understand the area of a triangle formula, which is really just half of a rectangle. So when we go back in this conceptual idea that I do want to get to this place where it’s like I know and trust the formula for area of a trapezoid. But if, for whatever reason, I can’t find, I can’t remember this formula, it’s like well wait a second, what is a trapezoid. All right, let’s draw one. What other shapes… I know that the formula came from other shapes. What shapes can I break down? I can break it into two triangles or I could break it into two triangles and a rectangle. If I know in my soul that conceptually this came from somewhere then I can always figure it out. It might take me a little longer, but then it’s like having that confidence. You don’t have to give up because I forgot a formula because I had no understanding at all. So we want both of those things so that we can have that efficiency of time. It’s not the only goal, it’s just one of the benefits of having this full perspective when we talk about math proficiency. I’m glad that you referenced that and I’m glad that you see some of that value as well. So that’s great.

Michael Rubin: One of the things I thought about in terms of task, I think back to I always get that question. When am I ever going to use this, Mr. Rubin? And I always went to that idea of like well, how many of you play sports who plays football? Who lifts weights? All right, are you lifting weights there? And I realized what I was asking didn’t correlate with what I was doing in class. It was like saying, yeah in math you’re learning how to problem solve. You’re learning how to think. You’re learning how to make connections and you’re growing your brain so that it becomes easier to make those connections and your learning patterns of thinking. Because ultimately what I’ve always believed is that math was about problem solving and relationships and connections and observing patterns and figuring that out. But by teaching through procedure I was the one doing all that work. I was the one making the conceptual understanding and saying all that stuff. And then I’m like, okay now you do the procedures and you’re going to learn by osmosis.

Michael Rubin: So when you talk about teaching through tasks that’s where I think of this dichotomy between teaching through tasks and procedural. And so it puts that negative light on the procedural because to me I associate that with being a liar to my students. Saying, yeah you’re doing this because this is what math is and not being authentic with that. I like the reminder that ultimately, yeah, you get to those procedures but… and I know Kyle and Jon, you’ve talked about this with the difference between automaticity and memorization. This idea that if you do something enough conceptually then you’re going to start to understand how the pieces work and you will do it more efficiently because that’s a faster, easier way to do it. And by teaching conceptually we’re teaching those relationships. Because what’s a concept other than a relationship between two different things, two different quantities or two different ideas?

Michael Rubin: That if we’re teaching through tasks they’re getting to that procedure. But then when I tell them, hey we’re doing this to build your minds and to build your brains and identify relationships, make connections, and use them to solve problems. Not necessarily even in math, but just in your own life. That we’re going to come to the procedures. I know the content skills are in the background, but I’m really being more authentic in what I’m telling them. And my actions and what I’m having them do in class matches up more.

Kyle Pearce: And I think it’s really important to show… and you mentioned Joe Bowler about really being clear on what we value. If I’m showing that I value the procedure more than say some other method like modeling with area models or number lines or whatever strategy the student uses. If it’s like I’m celebrating this procedure like this student has arrived and these ones haven’t then I’m perpetuating that idea. Society constantly having this idea of constantly having the conceptual there and the procedural and helping students to get there and asking them specific questions to help elicit that thinking and help them move there, but not fixing them.

Kyle Pearce: Cathy Fosnot says that all the time. She doesn’t want to fix the mathematician. She’ll use their thinking. And she wants to also help them develop their thinking a little bit further through the questioning. The through the questioning and the conferring piece. So that, for me, is a huge pice.

Kyle Pearce: So it’s getting late here. I think this is going to definitely have to be a multiple episode podcast. Jon, what’s our last question for this series of questions for our good friend Michael here? What should we end this on before we bring Michael back on for another episode?

Jon Orr: Yeah, I think we should bring him back on because he’s got a list of questions here that he wants answered, and I think we definitely want to answer them. And I think they would all benefit everyone listening. But Michael, we’ve had a good chat about a various things this evening. We are wondering, what is your… if you think back to all those. What’s a big takeaway for you from this conversation? You mentioned a few things about the workshop and some of the learning you’ve done, those were big takeaways then. But what’s a big takeaway from this conversation here today?

Michael Rubin: Yeah, I’ve got a couple. I think especially getting to that question of how long I should wait before I spiral back to a topic. This idea of being flexible I think was my biggest takeaway. As I plan I should have a general idea to know that, yeah, it’s going to change. And I need to think about where the kids are at or if they’ve really got something then maybe you can take a little bit longer before returning to it. But if they’re struggling a little bit more with it maybe you come back to it a little sooner or you spend another day on it. That idea of pivoting instead of proceeding as planned. So that was definitely a big thing.

Michael Rubin: Can I give two takeaways?

Jon Orr: Sure.

Michael Rubin: So the second key takeaway I have is I really liked when Kyle was talking about purposeful practice about picking problems with intentionality. Meaning looking at what the student needs maybe both as a group and as the individuals and how when you pick the problems with intentionality you can also use that information as formative assessment to determine what the kids need. Because maybe there’s something they need that you weren’t anticipating that will help you make that decision about when to spiral back.

Kyle Pearce: Those are some deep takeaways. I think you’ve nailed it. On purposeful practice there for sure. We are looking forward to having you back on the show. We know that we can talk so much more about these ideas. And I think the people listening would love to hear that too. Michael, we want to thank you for joining us on the podcast. And we hope you enjoy the rest of your summer. And back to school, and I hope all goes well for you back to school in August. We’re recording right now in June. So only about a month left for your summer vacation. Hope you enjoy that.

Michael Rubin: Yeah, thank you very much.

Jon Orr: Thanks Michael. We’ll be in touch to get you hooked up for another time slot to come join us. We are really excited to carry on this conversation. Thanks a ton.

Michael Rubin: Thank you.

Kyle Pearce: As always, both Jon and I learned so much from these Math Mentoring Moment episodes. But in order to ensure we hang on to this new learning instead of letting it fade away like footprints in the sand we must reflect on what we’ve learned. An excellent way to ensure that learning sticks is to reflect and create a plan for yourself to take action on something that resonated with you. A great way to hold yourself accountable is to write down, and even better, share it with someone. Your partner, colleagues, or with the Math maker community by commenting on the show notes page, tagging @makemathmoments on social media, or in our free private Facebook group, Math Moment Makers Okay through 12.

Jon Orr: Another great way to 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 through our program we’re confident you too can create resilient, engaged problem solvers just like Michael has learned to do.

Kyle Pearce: And if taking a deep dive alone isn’t enough the workshop is accredited for two to four PD credit hours through Brandman University. If you’re interested in learning more about registering be sure to check out makemathmoments.com/onlineworkshop.

Jon Orr: If you want to join us on the podcast for an upcoming Math Mentoring Moment episode where you too can share a big math class struggle just like Michael did you can apply over at makemathmoments.com/mentor.

Kyle Pearce: In order to ensure you don’t miss out on any new episodes as they come out each week be sure to subscribe on your favorite podcast platform. And if you’re liking what you’re hearing do us a huge solid by sharing the podcast with a colleague and help us reach a wider audience by leaving us a review on iTunes, Spotify, Google Play or wherever you’re listening to this particular episode. Each and every subscribe helps.

Jon Orr: Shownotes and links to resources for this episode can be found at makemathmoments.com/episode40. Again, that is makemathmoments.com/episode40.

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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